SlideShare a Scribd company logo

Vector analysis

Download as PPT, PDF
Solo Hermelin, profile picture
Solo Hermelin

This document provides a table of contents for a textbook on vector analysis. The table of contents covers topics including: vector algebra, reciprocal sets of vectors, vector decomposition, scalar and vector fields, differential geometry, integration of vectors, and applications of vector analysis to fields such as electromagnetism and fluid mechanics. Many mathematical concepts are introduced, such as vector spaces, differential operators, vector differentiation and integration, and theorems relating concepts like divergence, curl and gradient.

Science
1 of 269
Downloaded 542 times
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Most read
30
31
32
33
34
35
36
Most read
37
38
39
40
41
42
43
44
Most read
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
1 VECTOR ANALYSIS SOLO HERMELIN http://www.solohermelin.com
2 Vector AnalysisSOLO TABLE OF CONTENT Algebras History Vector Analysis History Vector Algebra Reciprocal Sets of Vectors Vector Decomposition The Summation Convention The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee   Change of Vector Base, Coordinate Transformation Vector Space Differential Geometry Osculating Circle of C at P Theory of Curves Unit Tangent Vector of path C at a point P Curvature of curve C at P Osculating Plane of C at P Binormal Torsion Seret-Frenet Equations Involute Evolute Vector Identities Summary Cartesian Coordinates
3 Vector AnalysisSOLO TABLE OF CONTENT (continue – 1) Differential Geometry Conjugate Directions Surfaces in the Three Dimensional Spaces First Fundamental Form: Arc Length on a Path on the Surface Surface Area Change of Coordinates Second Fundamental Form Principal Curvatures and Directions Asymptotic Lines Scalar and Vector Fields Vector Differentiation Ordinary Derivative of Scalars and Vectors Partial Derivatives of Scalar and Vectors Differentials of Vectors The Vector Differential Operator Del (, Nabla) Scalar Differential Vector Differential Differential Identities
4 Vector AnalysisSOLO TABLE OF CONTENT (continue – 2) Scalar and Vector Fields Curvilinear Coordinates in a Three Dimensional Space Covariant and Contravariant Components of a Vector in Base .321 ,, uuu rrr  Coordinate Transformation in Curvilinear Coordinates Covariant Derivative Covariant Derivative of a Vector .A  Vector Integration Ordinary Integration of Vectors Line Integrals Surface Integrals Volume Integrals Simply and Multiply Connected Regions Green’s Theorem in the Plane Stoke’s Theorem Divergence Theorem Gauss’ Theorem Variations Stokes’ Theorem Variations Green’s Identities Derivation of Nabla (  ) from Gauss’ Theorem The Operator .
5 Vector AnalysisSOLO TABLE OF CONTENT (continue – 3) Scalar and Vector Fields Fundamental Theorem of Vector Analysis for a Bounded Region V (Helmholtz’s Theorem) Reynolds’ Transport Theorem Poisson’s Non-homogeneous Differential Equation Kirchhoff’s Solution of the Scalar Helmholtz Non-homogeneous Differential Equation Derivation of Nabla (  ) from Gauss’ Theorem The Operator . Orthogonal Curvilinear Coordinates in a Three Dimensional Space Vector Operations in Various Coordinate Systems Applications Laplace Fields Harmonic Functions Rotations
6 Synthetic Geometry Euclid 300BC Algebras HistorySOLO Extensive Algebra Grassmann 1844 Binary Algebra Boole 1854 Complex Algebra Wessel, Gauss 1798 Spin Algebra Pauli, Dirac 1928 Syncopated Algebra Diophantes 250AD Quaternions Hamilton 1843 Tensor Calculus Ricci 1890 Vector Calculus Gibbs 1881 Clifford Algebra Clifford 1878 Differential Forms E. Cartan 1908 First Printing 1482 http://modelingnts.la.asu.edu/html/evolution.html Geometric Algebra and Calculus Hestenes 1966 Matrix Algebra Cayley 1854 Determinants Sylvester 1878 Analytic Geometry Descartes 1637 Table of Content
7 Vector Analysis HistorySOLO John Wallis 1616-1703 1673 Caspar Wessel 1745-1818 “On the Analytic Representation of Direction; an Attempt”, 1799 bia  Jean Robert Argand 1768-1822 1806 1i Quaternions 1843 William Rowan Hamilton 1805-1865 3210 qkqjqiq  Extensive Algebra 1844 Herman Günter Grassmann 1809-1877 “Elements of Vector Analysis” 1881 Josiah Willard Gibbs 1839-1903 Oliver Heaviside 1850-1925 “Electromagnetic Theory” 1893 3. R.S. Elliott, “Electromagnetics”,pp.564-568 http://www-groups.dcs.st-and.ac.uk/~history/index.html Table of Content Edwin Bidwell Wilson 1879-1964 “Vector Analysis” 1901
8 Vector AnalysisSOLO ba   Vector Algebra b  a  a  Addition of Vectors Parallelogram Law of addition Subtraction of Vectors  baba  1 Parallelogram Law of subtraction b  a  b   b  a  ba   Multiplication of Vector by a Scalar am  a  a  am  b  a  ba   ba   b   Geometric Definition of a Vector A Vector is defined by it’s Magnitude and Directiona  a  a 
9 Vector AnalysisSOLO         bababa  ,sin b  a  ba  ,  b  a  ba  ,  ba   abba   Scalar product of two vectors ba  , Vector product of two vectors ba  ,   2/1 2/1 ,cos                   aaaaaaa Magnitude of Vector a  Unit Vector (Vector of Unit Magnitude)aa 1ˆ  a a aa   1 :1:ˆ  Zero Vector (Vector of Zero Magnitude)0  aa  0 0:0   00 a     ababbababa ba   ,cos, ,         Vector Algebra (continue – 1)
10 Vector AnalysisSOLO     n|| ˆˆˆˆ aanannana n    nˆ a   n a n a ˆ ˆ n||      n a n a n ˆ ˆ      Vector decomposition in two orthogonal directions nn ,|| Vector decomposition in two given directions (geometric solution) 1 ˆn a  2 ˆn A B C Given two directions and , and the vector a  1 ˆn 2 ˆn anBCnCA   21 ˆˆ 1 ˆn a  2 ˆn A B Draw lines parallel to those directions passing through both ends A and B of the vector . The vectors obtained are in the desired directions and by rule of vector addition satisfy a  Vector Algebra (continue – 2) Table of Content
11 SOLO Triple Scalar Product Vectors & Tensors in a 3D Space 3321 ,, Eeee   are three non-coplanar vectors, i.e. 1e  2e  3e      0:,, 321321  eeeeee                0,, ,,,, 123123213 132132132321   eeeeeeeee eeeeeeeeeeee   Reciprocal Sets of Vectors The sets of vectors and are called Reciprocal Sets or Systems of Vectors if: 321 ,, eee  321 ,, eee  DeltaKroneckertheis ji ji ee j i j i j i        1 0 Because is orthogonal to and then2e  3e 1 e         321 321321 1 132 1 ,, 1 ,,1 eee keeekeeekeeeeke    and in the same way and are given by:2 e  3 e 1 e             321 213 321 132 321 321 ,,,,,, eee ee e eee ee e eee ee e               
12 SOLO Vectors & Tensors in a 3D Space Reciprocal Sets of Vectors (continue) By using the previous equations we get:                321 3 2 321 13323132 2 321 133221 ,,,,,, eee e eee eeeeeeee eee eeee ee                        321 213 321 132 321 321 ,,,,,, eee ee e eee ee e eee ee e                        0 ,, 1 ,, ,, 321321 3 3321321    eeeeee ee eeeeee    Multiplying (scalar product) this equation by we get:3 e  In the same way we can show that: Therefore are also non-coplanar, and:321 ,, eee     1,,,, 321 321 eeeeee             321 21 3321 13 2321 32 1 ,,,,,, eee ee e eee ee e eee ee e                1e  2e  3e  1 e  2 e  3 e  Table of Content 1e  2e  3e 
13 SOLO Vectors & Tensors in a 3D Space Vector Decomposition Given we want to find the coefficients and such that:3 EA  321 ,, AAA 321 ,, AAA       3 1 3 3 2 2 1 1 3 1 3 3 2 2 1 1 j j j i i i eAeAeAeA eAeAeAeAA   3,2,1, iee i i  are two reciprocal vector bases Let multiply the first row of the decomposition by :j e  Let multiply the second row of the decomposition by :ie  j i j i i i j i ij AAeeAeA    3 1 3 1   i j i j j j i j ji AAeeAeA    3 1 3 1   Therefore: ii jj eAAeAA   & Then:                      3 1 3 3 2 2 1 1 3 1 3 3 2 2 1 1 j j j i i i eeAeeAeeAeeA eeAeeAeeAeeAA   Table of Content 1e  2e  3e 
14 SOLO Vectors & Tensors in a 3D Space The Summation Convention j j j j j eAeAeAeAeA    3 1 3 3 2 2 1 1 The last notation is called the summation convention, j is called the dummy index or the umbral index.             i i i i j j j j j j j j j i i i i i eAeeAeeAeeA eAeeAeeAeeAA         3 1 3 1 Instead of summation notation we shall use the shorter notation first adopted by Einstein  3 1j j j eA  j j eA  Table of Content
15 SOLO Vectors & Tensors in a 3D Space Let define: The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee   jiijjiij geeeeg   3321 ,, Eeee   the metric covariant tensors of By choosing we get:         j ijiii j jiiiii egegegeg eeeeeeeeeeeee     3 3 2 2 1 1 3 3 2 2 1 1 i eA   or: j iji ege   For i = 1, 2, 3 we have:                                                        3 2 1 332313 322212 312111 3 2 1 333231 232221 131211 3 2 1 e e e eeeeee eeeeee eeeeee e e e ggg ggg ggg e e e             1e  2e  3e 
16 SOLO Vectors & Tensors in a 3D Space We want to prove that the following determinant (g) is nonzero: The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee                            332313 322212 312111 333231 232221 131211 detdet: eeeeee eeeeee eeeeee ggg ggg ggg g    g is the Gram determinant of the vectors 321 ,, eee  Jorgen Gram 1850 - 1916 Proof: Because the vectors are non-coplanars the following equations:321 ,, eee  03 3 2 2 1 1  eee   is true if and only if 0321   Let multiply (scalar product) this equation, consecutively, by :321 ,, eee                                             0 0 0 0 0 0 3 2 1 332313 322212 312111 33 3 23 2 13 1 32 3 22 2 12 1 31 3 21 2 11 1       eeeeee eeeeee eeeeee eeeeee eeeeee eeeeee       Therefore α1= α2= α3=0 if and only if g:=det {gij}≠0 q.e.d.
17 SOLO Vectors & Tensors in a 3D Space Because g ≠ 0 we can take the inverse of gij and obtain: The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee   where Gij = minor gij having the following property: j i kj ik ggG                                                      3 2 1 333231 232221 131211 3 2 1 333231 232221 131211 3 2 1 1 e e e ggg ggg ggg e e e GGG GGG GGG g e e e          and: g G g gminor g ij ijij  Therefore: g g g g G gg j i kj ik kj ik   j i kj ik gg  Let multiply the equation by gij and perform the summation on ij iji ege   jj ij ij i ij eeggeg   Therefore: i ijj ege   Let multiply the equation byk kjj ege   i e  iji k kjijji geegeeee   jiijjiij geeeeg   i jkj ik ggG  The Operator .
18 SOLO Vectors & Tensors in a 3D Space The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee   Let find the relation between g and    321321 :,, eeeeee   We shall write the decomposition of in the vector base32 ee   321 ,, eee  3 3 2 2 1 1 32 eeeee    Let find λ1, λ2, λ3. Multiply the previous equation (scalar product) by .1e      i i ggggeeeeee 113 3 12 2 11 1 321321 ,,    Multiply this equation by g1i:   ii i ii ggeeeg     1 1 1321 1 ,, Therefore:  321 1 ,, eeeg ii   Let compute now:            321 1 0 323 3 0 322 2 321 1 3232 eeeeeeeeeeeeeeee                                321 11 321 11 321 2 233322 321 3322332 321 3232 321 32321 ,,,,,,,, ,,,, eee gg eee G eee ggg eee eeeeeee eee eeee eee eeee                From those equations we obtain:  321 11 1 ,, eee gg  Finally:     geeeeee  321 2 321 1 ,,,,   We can see that if are collinear than and g are zero.321 ,, eee  321 ,, eee  Table of Content
19 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation Let choose another base and its reciprocal 321 ,, fff   321 ,, fff                                                 3 2 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 e e e L e e e f f f ef j j ii               where j i j i ef   By tacking the inverse of those equations we obtain:                                                   3 2 1 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 f f f L f f f e e e fe i j ij               where j ij i ef   Because are the coefficients of the inverse matrix with coefficients :j i  j i i j i k k j  
20 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation (continue – 1) Let write any vector in those two bases:A                                                       3 2 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 e e e L e e e f f f ef ef ji j i j j ii                 then:                                                     3 2 1 1 3 2 1 3 3 3 2 3 1 2 3 2 2 2 1 1 3 1 2 1 1 3 2 1 E E E L E E E F F F ef EF T j ii j i j ji       i i j j fFeEA   iijj fAFeAE   & i j ji i i j j j j i i EFfEeEfF    or: But we remember that: We can see that the relation between the components F1, F2, F3 to E1, E2, E3 is not similar, contravariant, to the relation between the two bases of vectors to . Therefore we define F1, F2, F3 and E1, E2, E3 as the contravariant components of the bases and .  321 ,, fff   321 ,, eee   321 ,, fff   321 ,, eee  where
21 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation (continue – 2)                                              3 2 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 E E E L E E E F F F EF j iji     i i j j fFeEA   iijj fAFeAE   & then: Let write now the vector in the two bases andA   321 ,, fff   321 ,, eee  where and j ij ef ijjjjii EfeEeEAfAF j iji      We can see that the relation between the components F1, F2, F3 to E1, E2, E3 is similar, covariant, to the relation between the two bases of vectors to . Therefore wew define F1, F2, F3 and E1, E2, E3 as the covariant components of the bases and .  321 ,, fff   321 ,, eee   321 ,, fff   321 ,, eee 
22 SOLO Vectors & Tensors in a 3D Space We have: Change of Vector Base, Coordinate Transformation (continue – 3)     j j j j i i i i eeAeA eeAeAA     Ai contravariant component Aj covariant component Let find the relation between covariant and the contravariant components: j j j ij i ege i i eAegAeAA j iji     i i i ij j ege j j eAegAeAA i ijj     Therefore: ij j i ij i j gAAgAA  & Let find the relation between gij and gij defined in the bases and to and defined in the bases and . ie  i e  i f  i f  ij g ij g m m kkj j ii efef    & jm m k j imj m k j ikiik geeffg    Hence: jm m k j iik gg  This is a covariant relation of rank two, (similar, two times, to relation between to .i f  j e 
23 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation (continue – 4)                                                      3 2 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 e e e L e e e f f f ef ef ji j i j j ii                 Since we have:k iki fgf     m jm j i ege j j i ef k iki egefgf m jmjj j ii       and:     m jm j i km kjm j i gg k ik egfgfg mjm m k j iik     Therefore, by equalizing the terms that multiply we obtain:                                                  3 2 1 3 2 1 3 3 3 2 3 1 2 3 2 2 2 1 1 3 1 2 1 1 3 2 1 f f f L f f f e e e fe Tkm k m                jm j i g We found the relation:
24 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation (continue – 5) Therefore:                                                 3 2 1 1 3 2 1 3 3 3 2 3 1 2 3 2 2 2 1 1 3 1 2 1 1 3 2 1 e e e L e e e f f f ef Tmj m j               Let take the inverse of the relation by multiplying by and summarize on m:j m km k m fe   jkj k km k j m mj m fffe    From the relation: mk m kjj i i efef    & we have: jmk m j i mjk m j i kiik geeffg    or: jmk m j i ik gg  This is a contravariant relation of rank two. From the relation: m m kk jj i i efef    & we have: m j m k i jm jm k i j i kk i eeff    or: This is a relation once covariant and once contravariant of rank two. m j m k i j i k   Table of Content
25 SOLO Vectors & Tensors in a 3D Space Cartesian Coordinates Three dimensional cartesian coordinates are define as coordinates in a orthonormal basis such that:     zyxorkjieee 1,1,1ˆ,ˆ,ˆ,, 321   ix ˆ1  jy ˆ1  O kz ˆ1  x1 y1z1 0111111 1111111   zyzxyx zzyyxx yzxxyzzxy yxzxzyzyx 111111111 111111111       11111,1,12  zyxzyxg The reciprocal set is identical to the original set       ji ji ggg ij ijj iij 0 1  Given              z y x zyx A A A zyxzAyAxAA 111111               z y x A A A AMatrix notation of a vector
26 SOLO Vectors & Tensors in a 3D Space Cartesian Coordinates (continue – 1) ix ˆ1  jy ˆ1  O kz ˆ1  Given              z y x zyx A A A zyxzAyAxAA 111111               z y x A A A AMatrix notation of a vector              z y x zyx B B B zyxzByBxBB 111111                ABABBABABABA B B B AAAzByBxBzAyAxABA T zzyyxx T z y x zyxzyxzyx               111111              z y x B B B B
27 SOLO Vectors & Tensors in a 3D Space Cartesian Coordinates (continue – 2)           zyx zyxxyyxzxxzyzzy zyxzyx BBB AAA zyx zBABAyBABAxBABA zByBxBzAyAxABA 111 det111 111111             ABAB A A A BB BB BB BA B B B AA AA AA BA z y x xy xz yz z y x xy xz yz                                                   0 0 0 0 0 0 ix ˆ1  jy ˆ1  O kz ˆ1  Given              z y x zyx A A A zyxzAyAxAA 111111               z y x A A A AMatrix notation of a vector              z y x zyx B B B zyxzByBxBB 111111               z y x B B B B
28 SOLO Vectors & Tensors in a 3D Space Cartesian Coordinates (continue – 2) Table of Content                           ACBACBCBACBCBACBCB BACBACACBACACBACAC CBABACBABACBABA CCC BBB AAA BBB AAA CCC zCyCxC BBB AAA zyx CBA zxyyxyzxxzxyzzy zxyyxyzxxzxyzzy zxyyxyzxxzxyzzy zyx zyx zyx zyx zyx zyx zyx zyx zyx        detdet111 111 det                  ABCABCBAC B B B AA AA AA CCCBAC TT z y x xy xz yz zyx                           0 0 0 Given              z y x zyx A A A zyxzAyAxAA 111111               z y x A A A A Matrix notation of a vector              z y x zyx B B B zyxzByBxBB 111111               z y x B B B B              z y x zyx C C C zyxzCyCxCC 111111               z y x C C C C
29 Vector AnalysisSOLO            bacbacacbacbcbacba  ,,,,:,,        cbabcacba                    cbdadbca dcbcdba dcbadcba                  adcbbdca dcbacdbadcba   ,,,, ,,,,          2 ,, cbaaccbba                          feabdcfebadc dcefbadcfeba fedcbafecdbafedcba    ,,,,,,,, ,,,,,,,, ,,,,,,,,    Vector Identities Summary       0 bacacbcba  Table of Content
30 SOLO VECTOR SPACE Given the complex numbers .C ,, A Vector Space V (Linear Affine Space) with elements over C if its elements satisfy the following conditions: Vzyx   ,, I. Exists a operation of Addition with the following properties: xyyx   Commutative (Abelian) Law for Addition1    zyxzyx   Associative Law for Addition2 xx   0 Exists a unique vector0  3 II. Exists a operation of Multiplication by a Scalar with the following properties: 0..   yxtsVyVx4 Inverse xx  15    xx    Associative Law for Multiplication6   xxx    Distributive Law for Multiplication7   yxyx    Commutative Law for Multiplication8           00101010 3 575   xxxxxxxxWe can write: Vector Analysis
31 SOLO Scalar Product in a Vector Space The Scalar Product of two vectors is the operation with the symbol with the following properties: Vyx   ,   Cyx   ,    xyyx  ,,     yxyx  ,,        zyzxzyx  ,,,      00,&0,   xxxxx Distance Between Two Vectors The Distance between two Vectors is defined by the following properties: Vyx   ,   yxyxd  ,     00,&0,   xxxdxxd    xydyxd  ,,       yzdzxdyxd  ,,,  Vector Analysis Table of Content
32 SOLO Differential Geometry is the study of geometric figures using the methods of Calculus. Here we present the curves and surfaces embedded in a three dimensional space. Properties of curves and surfaces which depend only upon points close to a particular point of the figure are called local properties.. The study of local properties is called differential geometry in the small. Those properties which involve the entire geometric figure are called global properties. The study of global properties is called differential geometry in the large. Hyperboloid of RotationToroyd Mobius Movement Differential Geometry Differential Geometry in the 3D Euclidean Space Table of Content
33 SOLO Differential Geometry in the 3D Euclidean Space A curve C in a three dimensional space is defined by one parameter t,  tr   ur  rd P O a b C Theory of Curves Regular Parametric Representation of a Vector Function: parameter t, defined in the interval I and:   Ittrr  ,   tr  (i) is of class C1 (continuous and 1st order differentiable) in I Arc length differential:      td td rd td td rd td rd trdtrdsd          2/1 2/1 : We also can define      sdtrdtrdsd  2/1* :  (ii)   It td trd  0    Iinconstantnottr   Arc length as a parameter:  t t td td rd s 0  Regular Curves: A real valued function t = t (θ), on an interval Iθ, is an allowable change of parameter if: (i) t (θ) is of class C1 in Iθ (ii) d t/ d θ ≠ 0 for all θ in Iθ A representation on Is is a representation in terms of arc length or a natural representation  srr   Table of Content
34 SOLO Differential Geometry in the 3D Euclidean Space A curve C in a three dimensional space is defined by one parameter t,  tr   ur  rd P O a b C - arc length differential     td td rd td rd trdtrdsd 2/1 2/1 :          td rd td rd r sd rd t /::   - unit tangent vector of path C at P (tangent to C at P) 1x 2x 3x td rd r '  - tangent vector of path C at P (tangent to C at P) 0,0,sincos 321  baetbetaetar  Example: Circular Helix 0,0,cossin' 321  baebetaeta td rd r     2/122 2/1 ba td rd td rd td rd            321 2/122 cossin/: ebetaetaba td rd td rd t   Theory of Curves (continue – 1) We also can define      sdtrdtrdsd  2/1* :  t sd rd sd rd  * Unit Tangent Vector of path C at a point P Table of Content
35 SOLO Differential Geometry in the 3D Euclidean Space The earliest investigations by means of analysis were made by René Descartes in 1637.  tr   ur  rd P O a b C René Descartes 1596 - 1650 Pierre Fermat 1601 - 1665 Christian Huyghens 1629 - 1695 Gottfried Leibniz 1646 - 1716 The general concept of tangent was introduced in seventeenth century, in connexion with the basic concepts of calculus. Fermat, Descartes and Huyghens made important contributions to the tangent problem, and a complete solution was given by Leibniz in 1677. The first analytical representation of a tangent was given by Monge in 1785. Gaspard Monge 1746 - 1818 Theory of Curves (continue – 2)
36 SOLO Differential Geometry in the 3D Euclidean Space A curve C in a three dimensional space is defined by one parameter t,  tr  - arc length differential     td td rd td rd trdtrdsd 2/1 2/1 :          '/'/:: rr td rd td rd r sd rd t   - unit tangent vector of path C at P (tangent to C at P) Normal Plane to at P:t    00  trr  We also can define - arc length differential     sdtrdtrdsd  2/1* :  t sd rd sd rd  * O a C t  P r  b 0 r  NormalPlane   00  trr  Theory of Curves (continue – 3) Return to Table of Contents
37 SOLO Differential Geometry in the 3D Euclidean Space O a C t  P r  b NormalPlane   00  trr  0 r  Curvature of curve C at P: rt sd td k   : Since 01  tkttt sd td tt    Define nnkkkkkkn    1 /1:&/:  ρ – radius of curvature of C at P k – curvature of C at P A point on C where k = 0 is called a point of inflection and the radius of curvature ρ is infinite. '' st td sd sd rd td rd r           "'"'"' ' ''''' 22 stskstststs td sd sd td td sd ts td td st td d r td d r           32 '"''''' skntstskstrr   '' sr   3 1 '''' skntrr    3 ' ''' r rr k     Let compute k as a function of and :'r  ''r  Theory of Curves (continue – 4)
38 SOLO Differential Geometry in the 3D Euclidean Space 1x 2x 3x t  k  0,0,sincos 321  baetbetaetar  Example 2: Circular Helix 0,0,cossin' 321  baebetaeta td rd r       2/1222/122 2/1 bardsdba td rd td rd td rd             321 2/122 cossin/: ebetaetaba td rd td rd t    2122 sincos/ etet ba a td sd td rd t sd td k       1 x 2x 3 x t  k  0,sincos 21  aetaetar  Example 1: Circular Curve 0,cossin' 21  aetaeta td rd r       2/1222/122 2/1 bardsdba td rd td rd td rd           21 cossin/: etaetaa td rd td rd t    21 sincos 1 / etet atd sd td rd t sd td k     Theory of Curves (continue – 5) Table of Content
39 SOLO Differential Geometry in the 3D Euclidean Space O a C t  P b ntk   1  NormalPlane Osculating Plane   00  trr  0r    00  ktrr Osculating Plane of C at P is the plane that contains and P:     00  ktrr kt  , The name “osculating plane” was introduced by D’Amondans Charles de Tinseau (1748-1822) in 1780. O a C t  P b ntk   1  NormalPlane Osculating Plane   00  trr  0r    00  ktrr  The osculating plane can be also defined as the limiting position of a plane passing through three neighboring points on the curve as the points approach the given point. If the curvature k is zero along a curve C then: tarrconstartt   0 0 The curve C is a straight line. Conversely if C is a straight line: 0//0  tkaa td rd td rd ttarr  C a regular curve of class ≥2 (Cclass) is a straight line if and only if k = 0 on C Theory of Curves (continue – 6) Table of Content
40 SOLO Differential Geometry in the 3D Euclidean Space Osculating Circleof C at P is the plane that contains and Pkt  , Theory of Curves (continue – 6) The osculating circle of a curve C at a given point P is the circle that has the same tangent as C at point P as well as the same curvature. Just as the tangent line is the line best approximating a curve at a point P, the osculating circle is the best circle that approximates the curve at P. http://mathworld.wolfram.com/OsculatingCircle.html O a C t  P b ntk   1  Normal Plane Osculating Plane   00  trr  0r    00  ktrr  Osculating Circle Osculating Circles on the Deltoid The word "osculate" means "to kiss."
41 SOLO Differential Geometry in the 3D Euclidean Space Osculating Circleof C at P is the plane that contains and P kt  , Theory of Curves (continue – 6a) O a C t  P b ntk   1  Normal Plane Osculating Plane   00  trr  0r    00  ktrr  Osculating Circle 3 xy  xy /1 xy cos xy sin http://curvebank.calstatela.edu/osculating/osculating.htm xy tan Table of Content
42 SOLO Differential Geometry in the 3D Euclidean Space O a C t  P b ntk   1  NormalPlane Osculating Plane   00  trr  0r    00  ktrr b  Rectifying Plane   00  krr  Binormal ntb  : Tangent Line: Principal Normal Line: Binormal Line: Normal Plane: Rectifying Plane: Osculating Plane: tmrr   0 nmrr   0 bmrr   0   00  trr    00  nrr    00  brr  The name binormal was introduced by Saint-Venant Jean Claude Saint-Venant 1797 - 1886 Fundamental Planes:Fundamental Lines: Theory of Curves (continue – 7) Table of Content
43 SOLO Differential Geometry in the 3D Euclidean Space Torsion Suppose that is a regular curve of class ≥ 3 (Cclass) along which is of class C1. then let differentiate to obtain:  srr    sn       snstsb                        snstsnstsnsnksnstsnstsb   Since                 001  snsnsnsnsnsnsnsn  Therefore is normal to , meaning that is in the rectifying plane, or that is a linear combination of and . n n t  b           sbsstssn                        snssbsstsstsnstsb    O a C t  P b n  0r  b  The continuous function τ (s) is called the second curvature or torsion of C at P.      snsbs   Theory of Curves (continue – 8)
44 SOLO Differential Geometry in the 3D Euclidean Space Torsion (continue – 1) Suppose that the torsion vanishes identically (τ ≡0) along a curve , then srr           0 0 bsbsnssb    O a C t  P b n  0r  0 b  Since and are orthogonal st   sb          constbsrbtbsr sd d bsr sd d  0000 0  Therefore is a planar curve confined to the plane srr     constbsr  0  C a regular curve of class ≥3 (Cclass) is a planar curve if and only if τ = 0 on C 1x 2x 3x t  k  0,0,sincos 321  baetbetaetar  Example 2: Circular Helix    321 2/122 cossin ebetaetabat    21 sincos etetn            321 2/122 21321 2/122 cossin sincoscossin eaetbetbba etetebetaetabantb          21 1222/122 sincos etbetbbaba td bd sd td td bd sd bd b      122   babnb   Theory of Curves (continue – 9)
45 SOLO Differential Geometry in the 3D Euclidean Space Torsion (continue – 2) Let compute τ as a function of and :'',' rr  '''r  ttr sd td td rd sd rd r      '      tbknkttrtrtr sd d trtr sd d r    2 "''''       tkbkbknkbktnkbktbktbkbk trttrtrtrttrttrtrtrtr sd d r      2 332 '''"3''''"2"'"'                                        2 0 3 1 2 0 26 6 0 3 0 4 0 22 5232 32 ,,,,,,'''",' '''",'',",'''',','",','3 '''"'"''''"'3' '''"3'"'',, ktntkbntknntkktkbknknktrrrt rrrtrrrttrrrttrrrtt rrtrrttrrttrrtttr trttrtrtrtrtrrrr                    ' 1 / 1 rtdsdsd td t   3 ' ''' r rr k    We also found:      6 2 2 6 ' ''' ' '''",' ,, r rr k r rrr rrr            2 ''' '''",' rr rrr      Theory of Curves (continue – 10) Table of Content
46 SOLO Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations Theory of Curves (continue – 11) We found and      snssb       snskst   Let differentiate      stsbsn                                stsksbssnsbskstsnsstsbstsbsn     We obtain          sbsnskstst  00             sbsbsnstsksn   0          sbsnsstsb  00   or                                                        sb sn st s ssk sk sb sn st       00 0 00   Jean Frédéric Frenet 1816 - 1900 Those are the Serret – Frenet Equations of a curve. Joseph Alfred Serret 1819 - 1885
47 SOLO Let compute: Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations (continue – 1) Theory of Curves (continue – 12) Let show that if two curves C and C* have the same curvature k (s) = k* (s) and torsion τ (s) = τ*(s) for all s then C and C* are the same except for they position in space. Assume that at some s0 the triads and coincide.      999 ,, sbsnst       999 *,*,* sbsnst     ******** * nttnknkttnktttttt sd d kk             ************ * * bnnbnttnkbtknnbtknnnnnn sd d kk          ******** * nbbnnbbnbbbbbb sd d      Adding the equations, we obtain:   0***  bbnntt sd d  Integrating we obtain:     30 ******  sbbnnttconstbbnntt  Since: and1,,1 ***  bbnntt    3***  bbnntt  we obtain: 1***  bbnntt  Finally since:         constsrsr sd rd stst sd rd  * * *    
48 SOLO Existence Theorem for Curves Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations (continue – 2) Theory of Curves (continue – 13) Let k (s) and τ (s) be continuous functions of a real variable s for s0 ≤ s ≤ sf. Then there exists a curve , s0 ≤ s ≤ sf, of class C2 for which k is the curvature, τ is the torsion and s is a natural parameter.  srr   332211332211332211 ,, ebebebbenenennetetett          tnktttttt sd d   2          nbntknnnnnn sd d   22    bnbbbbbb sd d               2 with: Proof: Consider the system of nine scalar differential equations:                       3,2,1,,,  isnssbsbsstsksnsnskst iiiiiii   and initial conditions:       302010 ,, esbesnest              btttktnkntntnt sd d             nnbbbtkbnbnbn sd d                  ntbnkbtbtbt sd d          and initial conditions:             1,0,1,0,0,1 000000  ssssss bbbnnnbtnttt 
49 SOLO Existence Theorem for Curves (continue – 1) Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations (continue – 3) Theory of Curves (continue – 14)                                    bnbb sd d ntbnkbt sd d nnbbbtkbn sd d btttktnknt sd d nbntknn sd d tnktt sd d          2 222 Proof (continue – 1): and initial conditions:             1,0,1,0,0,1 000000  ssssss bbbnnnbtnttt  We obtain: The solution of this type of differential equations with given initial conditions has a unique solution and since is a solution, it is unique.             1,0,1,0,0,1  bbbnnnbtnttt  The solution is an orthonormal triad.bnt  ,, We now define the curve:     s s dtsrr 0 :   We have: and , therefore k (s) is the curvature.1 tr          1&  snsnskst  Finally since:       nbtttknnkntntbntb    Therefore τ (s) is the torsion of  srr   q.e.d.
50 SOLO From the previous development we can state the following theorems: Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations (continue – 4) Theory of Curves (continue – 15) A curve is defined uniquely by the curvature and torsion as functions of a natural parameter. The equations k = k (s), τ = τ (s), which give the curvature and torsion of a curve as functions of s are called the natural or intrinsec equations of a curve, for they completely define the curve. O 0s C t  P n  0r  b  k 1  f s Fundamental Existence and Uniqueness Theorem of Space Curves Let k (s) and τ (s) be arbitrary continuous functions on s0≤s≤sf. Then there exists, for position in space, one and only one space curve C for which k (s) is the curvature, τ (s) is the torsion and s is a natural parameter along C. O 0s C t  P n  b  f s * C 0 r  * 0r  Table of Content
51 SOLO Let consider a space curve C. We construct the tangent lines to every point on C and define an involute Ci as any curve which is normal to every tangent of C. Differential Geometry in the 3D Euclidean Space Involute Theory of Curves (continue – 16) From the Figure we can see that the equation of the Involute is given by: turr  1 Differentiating this equation we obtain: 11 1 1 1 sd sd t sd ud nkut sd sd t sd ud sd td u sd rd t sd rd                  Scalar multiply this equation by and use the fact that and from the definition of involute : t  0nt  01 tt     1101 10 sd sd tt sd ud ntkutttt         01  sd ud scu         stscsrsr  1 C i C O r  1r  t  1 t  s c  Involute Curve
52 SOLO Differential Geometry in the 3D Euclidean Space Involute (continue – 1) Theory of Curves (continue – 17) C i C O r  1r  t  1 t  s c  Involute Curve        stscsrsr  1      n sd sd ksc sd sd t sd td sc sd rd sd rd t t     111 1 1             and are collinear unit vectors, therefore:1t  n     kscsd sd sd sd ksc   1 1 11 The curvature of the involute, k1, is obtained from:      ksc btk kscsd nd sd sd sd td nk sd td nt kscsd sd             11 1 1 1 1 11 1 1 Hence:   22 22 2 1 ksc k k     For a planar curve (τ=0) we have:   t sc nk     1 011 
53 SOLO Differential Geometry in the 3D Euclidean Space Involute (continue – 3) Theory of Curves (continue – 18) C i C O r  1r  t  1 t  s c  Involute Curve http://mathworld.wolfram.com/Involute.html Table of Content
54 SOLO The curve Ce whose tangents are perpendicular to a given curve C is called the evolute of the curve. Differential Geometry in the 3D Euclidean Space Evolute Theory of Curves (continue – 19) 11 twrbvnurr   Differentiating this equation we obtain:   11 1 1 1 sd sd b sd vd n sd ud nvbtkut sd sd b sd vd n sd ud sd bd v sd nd u sd rd t sd rd                      Scalar multiply this equation by and use the fact that and from the definition of evolute : t  0 btnt  01 tt    111 1 0 sd sd ttkutttt         01  ku  k u 1 C e C O r  1r  t 1 t  Evolute Curve The tangent to Ce, , must lie in the plane of and since it is perpendicular to . Therefore: n  b  t 1t  1 1 sd sd n sd ud vb sd vd ut                       
55 SOLO Differential Geometry in the 3D Euclidean Space Evolute (continue – 1) Theory of Curves (continue – 20)    ccuv   tantan  k u 1 C e C O r  1r  t 1 t  Evolute Curve We obtained: 1 1 sd sd n sd ud vb sd vd ut                            111 // wbvnuwrrt  But: Therefore: v v sd ud u u sd vd     or:                  u v sd d vu sd ud v sd vd u 1 22 tan c u v ds s s          1 tan 0  and:  bcnrr    tan1 We have one parameter family that describes the evolutes to the curve C.
56 SOLO Differential Geometry in the 3D Euclidean Space Evolute (continue – 2) Theory of Curves (continue – 21) http://math.la.asu.edu/~rich/MAT272/evolute/ellipselute.html Evolute of Ellipse Evolute of Logarithmic Spiral also a Logarithmic Spiral Evolute of Parabola Table of Content C e C O r  1r  t 1 t  Evolute Curve
57 SOLO Differential Geometry in the 3D Euclidean Space The vector defines a surface in E3  vur ,  vu vu rr rr N       vur ,   vdvudur  ,  rd 2 rd r udru  vdrv  d Nd P O  vudur ,               22 2 22 22 2 2 ,2 2 1 , 2 1 ,,, vdudOvdrvdudrudrvdrudr vdudOrdrdvurvdvudurvur vvvuuuvu     The vectors and define the tangent plane to the surface at point P. P u u r r      P v v r r      Define: Unit Normal Vector to the surface at P vu vu rr rr N     : First Fundamental Form:       2222 22: vdGvdudFudEvdrrvdudrrudrrrdrdI vvvuuu         0 2 0,0,00:                  GF FEforConditionSylvester FEGGE vd ud GF FE vdudrdrdI Surfaces in the Three Dimensional Spaces Table of Contents
58 SOLO Arc Length on a Path on the Surface:                 b a b a vuvu b a tdvdrudrvdrudrtd td rd td rd td td rd L 2/1 2/1                                                                           b a b a td td vd td ud GF FE td vd td ud td td vd G td vd td ud F td ud EL 2/1 2/1 22 2 Surface Area:  vur ,  rd udru  vdrv  d P O      vdudFGEvdud GE F GE vdud rr rr rrvdudrrrr vdudrrrrvdudrrvdrudrd vu vu vuvuvu vuvuvuvu 2 2/1 2 2/1 2 2/12 1 1,cos1 ,sin                                 vdudFGEd 2  vur ,  rd udru  vdrv P O a b Differential Geometry in the 3D Euclidean Space Table of Contents
59 SOLO Change of Coordinates  vur ,  rd udru  vdrv  d P O vdrv  udru  vdrudrvdrudrd vuvu   vdudFGEvdud vu vu JFGEvdudFGEd 222 , ,           vurvurr ,,  Change of coordinates from u,v to θ,φ    vuvv vuuu , ,  The coordinates are related by                   v u vv uu vd ud vu vu       I vd ud GF FE vdud vd ud vv uu GF FE vu vu vdud vd ud GF FE vdudI vu vu vv uu                                                      td td vd td ud GF FE td vd td ud td td vd td ud GF FE td vd td ud td td rd td rd Ld 2/12/1 2/1                                                                                                                         vu vu JFGE vv uu FGE vv uu GF FE vu vu GF FE FGE vu vu vu vu vv uu , , detdetdetdetdet 22 ** ** 2 Arc Length on a Path on the Surface and Surface Area are Invariant of the Coordinates: First Fundamental Form is Invariant to Coordinate Transformation Differential Geometry in the 3D Euclidean Space Table of Contents
60 SOLO vu vu rr rr N       vur ,   vdvudur  ,  rd 2 rd r udru  vdrv  d Nd P O  vudur ,  Second Fundamental Form: NdrdII :           22 2 2 2 2 : vdNvdudMudL vdNrvdudNrNrudNr vdNudNvdrudrNdrdII N vv M uvvu L uu vuvu             vdNudNNdNNdNN vu   01                NrNrNrNrNr vd d NrNrNrNrNr ud d Nr vuvuvuvuu uuuuuuuuu u    0 0 0                NrNrNrNrNr vd d NrNrNrNrNr ud d Nr vvvvvvvvv vuuvuvvuv v    0 0 0 Differential Geometry in the 3D Euclidean Space
61 SOLO vu vu rr rr N       vur ,   vdvudur  ,  rd 2 rd r udru  vdrv  d Nd P O  vudur ,  Second Fundamental Form: NdrdII :       2 2 2 : vdNrvdudNrNrudNrNdrdII N vv M uvvu L uu          NrNr uuuu   NrNr vuuv         NrNr NrL uuuu uu            uvvu vuuv vuvu NrNrM NrNr NrNr          NrNr NrN vvvv vv   NrNr vuvu   NrNr vvvv   22 2: vdNvdudMudLNdrdII  NrL uu   NrM vu   NrN vv   Differential Geometry in the 3D Euclidean Space
62 SOLO vu vu rr rr N       vur ,  O  vdvudur  , udru  vdrv  rd Second Fundamental Form: NdrdII :                33 3 3223 22 33 3 32 ,33 6 1 2 2 1 , 6 1 2 1 ,,, vdudOvdrvdudrvdudrudr vdrvdudrudrvdrudr vdudOrdrdrdvurvdvudurvur vvvvuvvuuuuu vvvuuuvu                 IINvdudOvdNvdudMudL NvdudOvdNrvdudNrudNr NvdudONrdNrdNrdNr vvvuuu 2 1 ,2 2 1 ,2 2 1 , 6 1 2 1 22 2 22 22 2 22 33 3 32 0         Differential Geometry in the 3D Euclidean Space
63 SOLO N  Second Fundamental Form: NdrdII : N  N  (i) Elliptic Case (ii) Hyperbolic Case (iii) Parabolic Case 02 MNL 02 MNL 0 &0 222 2   MNL MNL Differential Geometry in the 3D Euclidean Space
64 SOLO Differential Geometry in the 3D Euclidean Space (continue – 6a)  vur ,  vdrv  P O N  1nr  2nr  udru  2 M 1 M 02  MNL Dupin’s Indicatrix N  1n r  2n r  P 2 M 1 M 02  MNL N  1nr 2nr  P 1M 2M 0 0 222 2   MNL MNL http://www.mathcurve.com/surfaces/inicatrixdedupin/indicatrixdedupin.html Pierre Charles François Dupin 1784 - 1873 We want to investigate the curvature propertiesat a point P.     IINvdudOvdNvdudMudLNr 2 1 ,2 2 1 22 2 22   The expression 12 2 221 2 1  xNxxMxL was introduced by Charles Dupin in 1813 in “Développments de géométrie”, to describe the local properties of a surface. Second Fundamental Form: NdrdII : http://www.groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html Differential Geometry in the 3D Euclidean Space
65 SOLO N  Second Fundamental Form: NdrdII : N  (iv) Planar Case 0 MNL      3223 33 3 3223 6 1 ,33 6 1 vdDvdudCvdudBudA vdudOvdrNvdudrNvdudrNudrNNr vvvvuvvuuuuu    DxCxBxA  23 has 3 real roots Monkey Saddle DxCxBxA  23 has one real root Differential Geometry in the 3D Euclidean Space
66 SOLO Second Fundamental Form: NdrdII :    vurvurr ,,  Change of coordinates from u,v to θ,φ    vuvv vuuu , ,  The coordinates are related by                   v u vv uu vd ud vu vu          2222 22 uuuuuvvuuvuuuuuu vNvuMuLNvrvururNrL       vuvuvuvuvuvvvuvuvuuvvuuuvu vvNvuuvMuuLNvvrvuruvruurNrM            2222 2 vvvvvvvvvuvvvvuvuuvv vNvuMuLNvruvrvururNrN   Unit Normal Vector to the surface at P vu vu vu vu rr rr rr rr N          : uvuuvuu vrur u v r u u rr         vvvuvuv vrur v v r v u rr               II vd ud NM ML vdud vd ud vv uu NM ML vu vu vdud vd ud NM ML vdudII vu vu vv uu                                                      Second Fundamental Form is Invariant (unless the sign) to Coordinate Transformation Differential Geometry in the 3D Euclidean Space Table of Contents
67 SOLO N  Osculating Plane of C at P Principal Normal Line of C at P Surface t  P k  n1  vur ,  Normal Curvature - Length differential  2/1 rdrdrdsd      tvturr ,  Given a path on a surface of class Ck ( k ≥ 2) we define: td rd td rd sd rd t /:   - unit vector of path C at P (tangent to C at P) td rd td td sd td k /:   - curvature vector of path C at P       curvatureofradius nn nnk sd td k  111 1 1 1   NNkkn  : - normal curvature vector to C at P      /coscos1 :   kNnk Nkkn   - normal curvature to C at P Differential Geometry in the 3D Euclidean Space
68 SOLO N  Osculating Plane of C at P Principal Normal Line of C at P Surface t  P k  n1  vur ,  Normal Curvature (continue – 1) N Because C is on the surface, is on the tangent plan normal to . t    td Nd tN td td td Nd tN td td Nt td d Nt        00 and         vdrudrvdrudrvdNudNvdrudr td rd td rd td Nd td rd td rd td Nd td rd td rd td Nd t td rd N td td N sd td Nkk vuvuvuvu n                   / / /// 2 G vd ud F vd ud E N vd ud M vd ud L I II vdGvdudFudE vdNvdudMudL td vd G td vd td ud F td ud E td vd N td vd td ud M td ud L kn                                                                              2 2 2 2 2 2 2 2 22 22 22 22 Differential Geometry in the 3D Euclidean Space
69 SOLO Normal Curvature (continue – 2) G vd ud F vd ud E N vd ud M vd ud L I II vdGvdudFudE vdNvdudMudL td vd G td vd td ud F td ud E td vd N td vd td ud M td ud L kn                                                                              2 2 2 2 2 2 2 2 22 22 22 22 - kn is independent on dt therefore on C. - kn is a function of the surface parameters L, M, N, E, F, G and of the direction . vd ud - Because I = E du2 + 2 F du dv + G dv2 > 0 → sign kn=sign II - kn is independent on coordinates since I and II are independent.  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C Differential Geometry in the 3D Euclidean Space Table of Contents
70 SOLO Principal Curvatures and Directions G vd ud F vd ud E N vd ud M vd ud L I II vdGvdudFudE vdNvdudMudL kn                             2 2 2 2 2 2 22 22 - kn is a function of the surface parameters L, M, N, E, F, G and of the direction .vd ud Let find the maximum and minimum of kn as functions of the directions d u/ d v.  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C If this occurs for d u0/ d v0 we must have:         0&0 00 00 0000 00 00 , 2 ,, 2 ,           vdud vdvd vdud n vdud udud vdud n I IIIIII v k I IIIIII u k               0&0 00 00 00 00 00 00 00 00 00 00 , ,, 0 , ,, 0                   vdud vdnvd vdud vdvd vdud n vdud udnud vdud udud vdud n IkII I II III v k IkIII I II II u k Multiply by I and use  00 , 0 vdud n I II k  Differential Geometry in the 3D Euclidean Space
71 SOLO Principal Curvatures and Directions (continue – 1)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C           0&0 00 00 00 00 00 00 , , 0 , , 0       vdud vdnvd vdud n vdud udnud vdud n IkII v k IkII u k 22 2: vdNvdudMudLNdrdII  22 2: vdGvdudFudErdrdI  00 220 vdFudEI ud  00 220 vdGudFI vd  00 220 vdMudLII ud  00 220 vdNudMII vd       0 00 00 00 , , 0    vdud udnud vdud n IkII u k      0 00 00 00 , , 0    vdud vdnvd vdud n IkII v k     00000 0  vdFudEkvdMudL n     00000 0  vdGudFkvdNudM n Differential Geometry in the 3D Euclidean Space
72 SOLO We found: Principal Curvatures and Directions (continue – 2)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C              0 0 0000 0000 0 0 vdGudFkvdNudM vdFudEkvdMudL n n or:                       0 0 0 0 00 00 vd ud GkNFkM FkMEkL nn nn This equation has non-trivial solution if: 0det 00 00            GkNFkM FkMEkL nn nn or expending:       02 222 00  MNLkMFLGNEkFGE nn Differential Geometry in the 3D Euclidean Space
73 SOLO Study of the quadratic equation: Principal Curvatures and Directions (continue – 3)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C The discriminant of this equation is:       02 222 00  MNLkMFLGNEkFGE nn     222 42 MNLFGEMFLGNE      2 22 222 2 2 2222 2 2 22222424                      E LF LG E LF MFLGNEENLLFMELF E FGE LFMELFME E FGE     NLFNLGE E MLF LMGF E LF E LGF LFME E F LGNELFME E FGE 2 3 2 24222 2 2 2 44884424               E LGF LG E LGF LMGFLG NLGE E LF E MLF E LGF NLF E LF 22 22 22 22 2 24322 2 2 24 84884 488444            024 42 2 2 2 0 2 222                      LFME E F LGNELFME E FGE MNLFGEMFLGNE  Differential Geometry in the 3D Euclidean Space
74 SOLO Study of the quadratic equation (continue – 1): Principal Curvatures and Directions (continue – 4)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C The discriminant of this equation is:       02 222 00  MNLkMFLGNEkFGE nn          024 42 2 2 2 0 2 222                      LFME E F LGNELFME E FGE MNLFGEMFLGNE  The discriminant is greater or equal to zero, therefore we always obtain two real solutions that give extremum for kn: 21 , nn kk Those two solutions are called Principal Curvatures and the corresponding two directions are called Principal Directions    2211 ,,, vdudvdud 0&0  LGNELFME G N F M E L  The discriminant can be zero if:     02&0  LFME E F LGNELFME In this case: G N F M E L vdGvdudFudE vdNvdudMudL kn     22 22 2 2 This point in which kn is constant in all directions is called an Umbilical Point. Differential Geometry in the 3D Euclidean Space
75 SOLO Gaussian and Mean Curvatures Principal Curvatures and Directions (continue – 5)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C Rewrite the equation:       02 222 00  MNLkMFLGNEkFGE nn as:         0 2 2 2 2 2 00        FGE MNL k FGE MFLGNE k nn We define:    2 2 : 21 FGE MFLGNE kkH nn       2 2 21 : FGE MNL kkK nn    Mean Curvature Gaussian Curvature Karl Friederich Gauss 1777-1855 Differential Geometry in the 3D Euclidean Space
76 SOLO Gaussian and Mean Curvatures (continue – 1) Principal Curvatures and Directions (continue – 6)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C    2 2 21 : FGE MNL kkK nn    Gaussian Curvature    vurvurr ,,  Change of coordinates from u,v to θ,φ    vuvv vuuu , ,  The coordinates are related by                   v u vv uu vd ud vu vu     II vd ud NM ML vdud vd ud vv uu NM ML vu vu vdudII vu vu vv uu                                        We found:     I vd ud GF FE vdud vd ud vv uu GF FE vu vu vdudI vu vu vv uu                                                                  vu vu vv uu vv uu GF FE vu vu GF FE                           vu vu vv uu vv uu NM ML vu vu NM ML   2 2 2 2 detdetdetdet                                             vu vu vu vu vv uu FGE vv uu GF FE GF FE FGE   2 2 2 2 detdetdetdet                                             vu vu vu vu vv uu MNL vv uu NM ML NM ML MNL Therefore: invariant to coordinate changes        2 2 2 2 21 : FGE MNL FGE MNL kkK nn       Differential Geometry in the 3D Euclidean Space
77 SOLO Principal Curvatures and Directions (continue – 7)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2CStart with:              0 0 0000 0000 0 0 vdGudFkvdNudM vdFudEkvdMudL n n rewritten as :                       0 01 00000 0000 n kvdGudFvdNudM vdFudEvdMudL that has a nontrivial solution (1,-kn0) only if: 0det 0000 0000         vdGudFvdNudM vdFudEvdMudL or:       0 2 000 2 0  vdNFMGvdudNEGLudMEFL or:       0 0 0 2 0 0                  NFMG vd ud NEGL vd ud MEFL Differential Geometry in the 3D Euclidean Space
78 SOLO Principal Curvatures and Directions (continue – 8)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C We obtained: This equation will define the two Principal Directions 2211 21 & vdrudrrvdrudrr vunvun                  021 21 2 2 1 1 2 2 1 1 2112212121                              vdvdG MEFL NEGL F MEFL NFMG E vdvdG vd ud vd ud F vd ud vd ud E vdvdrrvdudvdudrrududrrrr vVvuuunn        0 0 0 2 0 0                  NFMG vd ud NEGL vd ud MEFL From the equation above we have:        MEFL NFMG vd ud vd ud MEFL NEGL vd ud vd ud       2 2 1 1 2 2 1 1 Let compute the scalar product of the Principal Direction Vectors: The Principal Direction Vectors are perpendicular. Differential Geometry in the 3D Euclidean Space
79 SOLO Principal Curvatures and Directions (continue – 9)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C Since the two Principal Directions are orthogonal 21 21 & vdrrudrr vnun   they must satisfy the equation: Let perform a coordinate transformation to the Principal Direction:  vu,       0 2 000 2 0  vdNFMGvdudNEGLudMEFL    21 ,0&0, vdud or:   0 2 1  udMEFL   0 2 2  vdNFMG 0 NFMG 01 ud 0 MEFL 02 vd 0E 0G 0 0   NrM rrF vu vu   at P Definition: A Line of Curvature is a curve whose tangent at any point has a direction coinciding with a principal direction at that point. The lines of curvature are obtained by solving the previous differential equation Differential Geometry in the 3D Euclidean Space
80 SOLO Principal Curvatures and Directions (continue – 10)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C Suppose (du0,dv0) is a Principal Direction, then they must satisfy the equations: Rodriguez Formula NrNrL uuuu   NrNrNrM vuuvvu   NrNrN vvvv                0 0 0000 0000 0 0 vdGudFkvdNudM vdFudEkvdMudL n n              0 0 0000 0000 0 0 vdrrudrrkvdNrudNr vdrrudrrkvdNrudNr vvvunvvuv vuuunvuuu   uu rrE   vu rrF   vv rrG                  0 0 0000 0000 0 0 vvunvu uvunvu rvdrudrkvdNudN rvdrudrkvdNudN            0 0 0 0 vn un rrdkNd rrdkNd   But are in the tangent plane at P since and are, and the vectors and are independent, therefore: rdkNd n  0  Nd  rd  vr  ur  00   rdkNd n The direction (du0,dv0) is a Principal Direction on a point on a surface if and only if from some scalar k, and satisfy:00 vdNudNNd vu   00 vdrudrrd vu   rdkNd   Rodriguez Formula We found: Differential Geometry in the 3D Euclidean Space Table of Contents
81 SOLO Conjugate Directions  vur ,  rd udru  vdrv P O N  Q NdN   l Let P (u,v) and Q (u+du,v+dv) neighboring points on a surface. The tangent planes to the surface at p and Q intersect along a straight line L. Now let Q approach P along a given direction (du/ dv=const= PQ), then the line l will approach a limit LC. The directions PQ and LC are called Conjugate Directions. Let be the normal at P and the normal at Q.N  NdN   Let the direction of LC be given by: vrurr vu    Since LC is in both tangential planes at P and at Q we have:   0&0  NdNrNr       0 vdNudNvrurNdr vuvu           0 vdvNrvduNrudvNruduNr vvvuuvuu   We found vvuvvuuu NrNNrNrMNrL   && The previous relation becomes:   0 vdvNvduudvMuduL  Given (du,dv) there is only one conjugate direction (δu,δv) given by the previous equation. Differential Geometry in the 3D Euclidean Space Table of Contents
82 SOLO Asymptotic Lines The directions which are self-conjugate are called asymptotic directions. becomes:   0 vdvNvduudvMuduL  We see that the asymptotic directions are those for which the second fundamental form vanishes. Moreover, the normal curvature kn vanishes for this direction. Those curves whose tangents are asymptotic directions are called asymptotic lines. v u vd ud   If a direction (du,dv) is self-conjugate than and the equation of conjugate lines 02 22  vdNvdudMudL The conjugat and asymptotic lines were introduced by Charles Dupin in 1813 in “Dévelopments de Géométrie”. Pierre Charles François Dupin 1784 - 1873 http://www.groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html Differential Geometry in the 3D Euclidean Space Table of Contents
83 SOLO Vectors & Tensors in a 3D Space Scalar and Vector Fields Let express the cartesian coordinates (x, y, z) of any point, in a three dimensional space as a function of three curvilinear coordinates (u1, u2, u3), where:  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3                 zyxuuzyxuuzyxuu uuuzuuuyuuuxx ,,,,,,,, ,,,,,,,, 332211 321321321 Those functions are single valued with continuous derivatives and the correspondence between (x,y,z) and (u1,u2,u3) is unique (isomorphism). kzjyixr   or 3 3 2 2 1 1 ud u r ud u r ud u r rd               321 ,,,, uuuzyx     321 ,,,, uuuAzyxAA   Assume now that the scalars Φ and vectors are functions of local coordinates, cartesian (x,y,z) or general, curvilinear (u1,u2,u3) A  In general ( we can not assume that Φ and are functions of position). A     rAAr   , Table of Contents
84 SOLO Vectors & Tensors in a 3D Space Vector Differentiation  tAA  Let a vector function of a single parameter tA  Ordinary Derivative of Scalars and Vectors The Ordinary Derivative of the Vector is defined as       t tAttA td tAd t      0 lim  tA   t t A     tAttAA   If the limit exists we say that is continuous and differentiable in t. tA  Differentiation Formulas If are differentiable vector functions of a scalar t and φ is a differentiable scalar of t, then CBA  ,,   td Bd td Ad BA td d      td Bd AB td Ad BA td d        td Ad A td d A td d         td Cd BAC td Bd ACB td Ad CBA td d          td Bd AB td Ad BA td d                            td Cd BAC td Bd ACB td Ad CBA td d       Table of Contents
85 SOLO Vectors & Tensors in a 3D Space Vector Differentiation Partial Derivatives of Scalar and Vectors    321 ,,,, uuuzyx     321 ,,,, uuuAzyxAA   Assume now that the scalars Φ and vectors are functions of local coordinates, cartesian (x,y,z) or general, curvilinear (u1,u2,u3) A  The partial derivatives are defined as follows       1 3213211 0 1 321 ,,,, lim ,, 1 u uuuuuuu u uuu u              2 3213221 0 2 321 ,,,, lim ,, 2 u uuuuuuu u uuu u              3 3213321 0 3 321 ,,,, lim ,, 3 u uuuuuuu u uuu u              1 3213211 0 1 321 ,,,, lim ,, 1 u uuuAuuuuA u uuuA u              2 3213221 0 2 321 ,,,, lim ,, 2 u uuuAuuuuA u uuuA u              3 3213321 0 3 321 ,,,, lim ,, 3 u uuuAuuuuA u uuuA u        Higher derivatives are also defined                                                                                           2 3 2 1 2 31 3 1212 2 2121 2 33 2 3 2 22 2 2 2 11 2 1 2 && && u A uuu A u A uuu A u A uuu A u A uu A u A uu A u A uu A   Table of Contents
86 SOLO Vectors & Tensors in a 3D Space Vector Differentiation Differentials of Vectors 3 3 2 2 1 1 ud u A ud u A ud u A zd z A yd y A xd x A Ad                     If     321321 111111,,,, 321 uAuAuAzAyAxAuuuAzyxAA uuuzyx        321321 111111111 321321 udAudAudAuAduAduAdzAdyAdxAdAd uuuuuuzyx     BdABAdBAd     BdABAdBAd     CdBACBdACBAdCBAd           CdBACBdACBAdCBAd   then If are differentiable vector functions of a scalar t.CBA  ,, Table of Contents
87 SOLO Vectors & Tensors in a 3D Space The Vector Differential Operator Del (, Nabla) We define the Vector Differential Operator Del (, Nabla) in Cartesian Coordinates as: z z y y x x 111:          This operator has double properties: (a) of a vector, (b) of a differential Gradient: Nabla operates on a Scalar or Vector Field z z y y x x z z y y x x 111111:                          zz z A yz z A xz z A zy y A yy y A xy y A zx x A yx x A xx x A zAyAxAz z y y x x A zyx zyx zyx zyx 111111 111111 111111 111111:                                            a scalar a dyadic
88 SOLO Vectors & Tensors in a 3D Space The Vector Differential Operator Del (, Nabla) (continue) We define the Vector Differential Operator in Cartesian Coordinates as: z z y y x x 111:          This operator has double properties: (a) of a vector, (b) of a differential Divergence: Nabla performs a Scalar Product on a Vector Field   z A y A x A zAyAxAz z y y x x A zyx zyx                        111111:  Curl (Rotor): Nabla performs a Vector Product on a Vector Field   z y A x A y x A z A x z A y A AAA zyx zyx zAyAxAz z y y x x A xyzxyz zyx zyx 111 111 111111:                                                          Table of Contents
89 SOLO Vectors & Tensors in a 3D Space Scalar Differential Let find the differentials of:    321 ,,,, uuuzyx  3 3 2 2 1 1 ud u ud u ud u zd z yd y xd x d                     rdzzdyydxxdz z y y x x d                 111111 Since      zyxuuzyxuuzyxuu ,,,,,,,, 332211  We obtain rduudrduudrduud   332211 ,, rdu u u u u u ud u ud u ud u d                             3 3 2 2 1 1 3 3 2 2 1 1 Comparing with we obtainrdd                             3 3 2 2 1 13 3 2 2 1 1 u u u u u uu u u u u u Using the Gradient definition: z z y y x x 111:          or 3 3 2 2 1 1 : u u u u u u          in general curvilinear coordinates Table of Contents
90 SOLO Vectors & Tensors in a 3D Space Vector Differential Let find the differentials of:    321 ,,,, uuuAzyxAA   3 3 2 2 1 1 ud u A ud u A ud u A zd z A yd y A xd x A Ad                         ArdzAyAxA z zd y yd x xd zdz z A y z A x z A ydz y A y y A x y A xdz x A y x A x x A Ad zyx zyxzyx zyx                                                              111 111111 111   ArdA u u u u u urdrdu u A u u A u u A Ad                                      3 3 2 2 1 13 3 2 2 1 1 rduudrduudrduud   332211 ,, and 3 3 2 2 1 1 : u u u u u u            ArdAd  Therefore In Cartesian Coordinates: In General Curvilinear Coordinates using Table of Contents
91 Vector AnalysisSOLO     Linearity of operator          Differentiability of operator   BABA   Linearity of operator   BABA   Linearity of operator   AAA    Differentiability of operator   AAA    Differentiability of operator Differential Identities
92 Vector AnalysisSOLO Differential Identities              BAAB BAAB BABABA BA BA                 AAA cbacabcba   2   0 0    aa      0 0 baabaa A      
93 Vector AnalysisSOLO Differential Identities          ABBAABBABA        BABABA B        ABBAAB A                ABBABABABAAB BA BA                  BAABABBABA        BABABA BA        ABABBA AAA        BABABA BBB  
94 Vector AnalysisSOLO         Differential Identities Summary   BABA     BABA     AAA      AAA             ABBAABBABA            BAABABBABA        BAABBA       AAA  2  0    0 A     AAAAA        2/ 2 Table of Contents
95 SOLO Vectors & Tensors in a 3D Space Curvilinear Coordinates in a Three Dimensional Space Let express the cartesiuan coordinates (x, y, z) of any point, in a three dimensional space as a function of three curvilinear coordinates (u1, u2, u3), where:  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3                 zyxuuzyxuuzyxuu uuuzuuuyuuuxx ,,,,,,,, ,,,,,,,, 332211 321321321 Those functions are single valued with continuous derivatives and the correspondence between (x,y,z) and (u1,u2,u3) is unique (isomorphism). kzjyixr   3 3 2 2 1 1 3 333 2 222 1 111 3 3 12 2 1 1 3 3 12 2 1 1 3 3 12 2 1 1 ud u r ud u r ud u r udk u z j u y i u x udk u z j u y i u x udk u z j u y i u x kud u z d u z ud u z jud u y d u y ud u y iud u x d u x ud u x kzdjydixdrd                                                                                                             or 3 3 2 2 1 1 ud u r ud u r ud u r rd           
96 SOLO Vectors & Tensors in a 3D Space Curvilinear Coordinates in a Three Dimensional Space (continue – 1) 3 3 2 2 1 1 ud u r ud u r ud u r rd            Let define: 3,2,1: 1     i u r r iu   If and are linear independent (i.e. if and only if αi = 0 i=1,2,3) then they form a base of the space E3. 21 , uu rr  3u r  0 3 1  i ui i r We have also:   3,2,1,,1  irdzyxuud i  We can write:         3 1 2 1 1 11 1 321 ,, udurudurudurrdzyxuud uuu   Because du1, du2, du3 are independent increments the precedent equation requires: 001 111 321  ururur uuu  Similarly by multiplying by and we obtain:rd  2 u 3 u         ji ji uru u r j i j u j i 0 1 1    Therefore and are reciprocal systems of vectors.321 ,, uuu rrr  321 ,, uuu   dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
97 SOLO Vectors & Tensors in a 3D Space Curvilinear Coordinates in a Three Dimensional Space (continue – 2) We proved that reciprocal systems of vectors are related by: and are reciprocal systems of vectors.321 ,, uuu rrr  321 ,, uuu       321 21 321 13 321 32 ,, , ,, , ,, 321 uuu uu uuu uu uuu uu rrr rr u rrr rr u rrr rr u                  321 21 321 13 321 32 ,, , ,, , ,, 321 uuu uu r uuu uu r uuu uu r uuu           and    1,,,, 321 321  uuurrr uuu  or 1 ,, ,, ,, ,, 321 321 333 222 111 333 222 111                                                    zyx uuu J uuu zyx J z u y u x u z u y u x u z u y u x u u z u y u x u z u y u x u z u y u x where is the Jacobian of x,y,z with respect to u1, u2, u3.         321 ,, ,, uuu zyx J Carl Gustav Jacob Jacobi 1804 - 1851  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
98 SOLO Vectors & Tensors in a 3D Space Curvilinear Coordinates in a Three Dimensional Space (continue – 3)   grrr u z u y u x u z u y u x u z u y u x uuu zyx J uuu                            321 ,,det: ,, ,, 333 222 111 321  If is nonsingular the transformation from x,y,z to u1, u2, u3 is unique.        321 ,, ,, uuu zyx J  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 g rr u g rr u g rr u uuuuuu 211332 321 ,,             211332 321 ,, uugruugruugr uuu   Table of Contents
99 SOLO Vectors & Tensors in a 3D Space Covariant and Contravariant Components of a Vector in Base .321 ,, uuu rrr  Given a vector we have: 321 ,, uuuA      j u j j u i u i uuu urAuAuAuAuA ruArArArArAA j ii     3 3 2 2 1 1 321 321 where: juj ii rAA uAA     : : are the contravariant components of A  are the covariant components of A  The Element of Arc ds. The Metric Coefficients gij Riemann and Euler Spaces Compute:       jiuujuiu ududrrudrudrrdrdsd jiji  2 Define: jiuuuuij grrrrg ijji   the metric coefficients jijijiij ududgududgrdrdsd  2 the element of arc  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 Leonhard Euler 1707- 1783 Georg Friedrich Bernhard Riemann 1826 - 1866A space with the metric defined above is called a Riemann Space. If gij = δij then the space is called an Euler Space.
100 SOLO Vectors & Tensors in a 3D Space Covariant and Contravariant Components of a Vector in Base (continue -1).321 ,, uuu rrr  or: If we substitute for we obtain:A  ju r    j ij j uu j ju ugurruAr jii                                      3 2 1 333231 232221 131211 3 2 1 u u u ggg ggg ggg r r r ugr u u u j ijui     Multiplying by gik (where gik gji = δj k) and summing on i and k: kjk j j ij ik u ik uuuggrg j    Changing k by j we obtain:                                    3 2 1 333231 232221 131211 3 2 1 u u u u ijj r r r ggg ggg ggg u u u rgu i      dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
101 SOLO Vectors & Tensors in a 3D Space Covariant and Contravariant Components of a Vector in Base (continue -2).321 ,, uuu rrr  Multiplying equation by we get:iu ijj rgu   i u   ij u iijji gruguu i    1 jiij uug or Now:       j ij j ij j ijui AguAgugArAA j   j iji AgA  We also found:    321321 ,,,,det 2 333231 232221 131211 uuuuuu rrrgrrr ggg ggg ggg g               dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 Table of Contents
102 SOLO Vectors & Tensors in a 3D Space Coordinate Transformation in Curvilinear Coordinates from those equations we obtain:                            32133 32122 32111 32133 32122 32111 ,, ,, ,, ,, ,, ,, uuuuu uuuuu uuuuu uuuuu uuuuu uuuuu Let and be two general curvilinear coordinates in an E3 space. There exists a unique transformation from one curvilinear coordinates to the other:  321 ,, uuu  321 ,, uuu k k j k k k j jj j i j j j i i ud u u ud u u udud u u ud u u ud               3 1 3 1 k k j j i j k k k j j i j j j i i ud u u u u ud u u u u ud u u ud                 3 1 3 1 3 1 Because dui and duk are independent:           ki ki u u u u i k k j j i 0 1  In the same way: k k i i j i i j j ud u u u u du u u ud         therefore:           kj kj u u u u j k k i i j 0 1   dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
103 SOLO Vectors & Tensors in a 3D Space Coordinate Transformation in Curvilinear Coordinates (continue – 1) If:    321321 ,,,, uuuruuurr   j i ij u u u r u r         ij u j i u r u u r     and: i j ji u u u r u r         ji u i j u r u u r     Let be a given vector with respect to two curvilinear coordinates.A  ji u j j ii u i rA u u ArAA          i j i ji u i j i u i ji u i i j j uA u u urA u u ur u u AurAuAuAA j ji                     i j ij A u u A    This is a contravariant relation with respect to the reference relation .ij u j i u r u u r     j i j i A u u A    This is a covariant relation with respect to the reference relation . ij u j i u r u u r      dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
104 SOLO Vectors & Tensors in a 3D Space Coordinate Transformation in Curvilinear Coordinates (continue – 2) Let define: j i j i i A u u AuA    11:  By multiplying the last equation by and because then and: 1     i j j i u u u u j i u u   j i j u u A    j j ij j i u u u uAuA    :  j j ii u u u u     From the reference relation mjki u j m uu i k u r u u rr u u r        & mkmkji uu j m i k u j m u i k uuij rr u u u u r u u r u u rrg              kl j m i k ij g u u u u g      This is a two order covariant relation with respect to the reference relation .ij u j i u r u u r     In the same way m m jjk k ii u u u uu u u u        & mk m j k im m jk k ijiij uu u u u u u u u u u u uug             km m j k iij g u u u u g      This is a two order contravariant relation with respect to the reference relation .ij u j i u r u u r      dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 Table of Contents
105 SOLO Vectors & Tensors in a 3D Space Covariant Derivative First we want to find the derivatives and . j u u r i    j i u u   Because are vectors of a base in E3 we can write as a function of this base. j u u r i    3,2,1ir iu  i i uij k ijj u r u r uu r                  :        ij k ij kEquivalent notation Where are the Cristoffel’s Symbols of II kind that we must determine.ij k  Elwin Bruno Cristoffel 1829 - 1900 Because thenk ji k uji k i u jiijj u uij k r u r u r uu r uu r r                                    ji k ij k  Let calculate now  mjigrrr u r mijkmij k uuij k u j u mkm i ,: ,      where are the Cristoffel’s Symbols of I kind  kmij k u j u mij gr u r mji m i       :,, Because andji k ij k  mkkm gg  kjikij ,, 
106 SOLO Vectors & Tensors in a 3D Space Covariant Derivative (continue – 1) To find let perform the following calculations:kij,    ijkjki k u uu k u uu kk ij u r rr u r rr uu g j ij i ji ,,                   ijkkji j u uu j u uu jj ik u r rr u r rr uu g k ik i ki ,,                   jikkji i u uu i u uu ii jk u r rr u r rr uu g k jk j kj ,,                 From those equations we obtain:                  k ij i jk j ik kij u g u g u g 2 1 ,                  k ij i jk j ik km kij km ij m u g u g u gg g 2 , Multiplying the equations by and summing we obtain:kmij m kij g , km g The Operator .
107 SOLO Vectors & Tensors in a 3D Space Covariant Derivative (continue – 2) Now let find . j i u u   Because are vectors of a base in E3 we can write as a function of this base: 3,2,1 iui j i u u   Tacking the derivative with respect to uj of the equation we get: k i k u ur i   0      j k u k j u u u ru u r i i   or ij kk mij mk uij mk j u j k u uru u r u u r m i i            k jk i j i u u u    : But we have also ij k ij k iuij k j k u ur u u r ii       Therefore: i ij k j k u u u    Because we have:mu imi rgu   iuij kim j k rg u u    
108 SOLO Vectors & Tensors in a 3D Space Covariant Derivative (continue – 3) We found that: m jkim i kmjmikjjki k ij gg u g    ,, Let find , where . k ij u g   jiij uug  j km iki km mjkj km ijmi km k j ij k i k ij gguuuu u u uu u u u g          j km iki km mj k ij gg u g    We can see that: 0       j kj i ki j jk i ki j km ik ij i km mj ij m jkim iji kmjm ij k ijij k ij ij k j m i j m i m gggggggg u g g u g g   This can be proven if we take the derivative with respect to uk of the equation: 1ij ij gg Table of Contents
109 SOLO or Vectors & Tensors in a 3D Space Covariant Derivative of a Vector .A  j ju i uArAA i   iimi i i u i mk i u k i mi u m ik i u k i k ui u k i k rAr u A rAr u A u r Ar u A u A                       ji mk i k i iju i mk i k i k uA u A grA u A u A i                            But we have also or ji mk i k i ij jm jkj k j k uA u A guA u A u A                           jm jkj j k j jm mj mkj j k j k j j j k j k uAu u A uAu u A u u Au u A u A                   Therefore we can write:               i mk i k i ij m jkj k j A u A gA u A By multiplying the equation by gij and summing we obtain:               m jkj k jiji mk i k i A u A gA u A Table of Contents
110 Vector AnalysisSOLO Dyadic Identities Summary       CbaCabCba        CbaCabCba     CCC      CCC        CCC  2  0 C  aCCa T    TT aCCa       BCaBaC TT  
111 SOLO Vector Integration Vector Analysis Ordinary Integration of Vectors Let be a vector depending on the single scalar variable t, with Ax (t), Ay (t), Az (t) continuous in a specific interval, then         ztAytAxtAtA zyx 111           zdttAydttAxdttAdttA zyx 111    If there exists a vector such that then:   tS td d tA   tS        ctSdttS td d dttA    where is an arbitrary constant.c  The definite integral between t = a and t = b gives           aSbSctSdttS td d dttA b a b a b a    Table of Contents
112 SOLO Vector Integration                         n i iiii n i iiiiin trzyxAtrtrzyxAS 11 1 ,,,,   C 1t n tb  2 t 0 ta  1i t i t 1 2 i n  itr  Let subdivide C into n parts by n arbitrary points t1, t2,…,tn, and call a=t0 and b=tn. On each arc joining ti-1 to zi choose a point ξi. Define the sum:                 C b a n i iiii z n n rdzyxArdzyxAtrzyxAS i  ,,,,,,limlim 1 0  Properties of Integrals     CCC rdBrdArdBA  constantrdArdA CC       a b b a rdArdA    b c c a b a rdArdArdA  Vector Analysis Line Integrals Let be continuous at all points on a curve C of a finite length L, defined by the position vector . tr  zyxA ,,  Let the number of subdivisions n increase in such a way that the largest of approaches zero, then the sum approaches a limit that is called the line integral (also Riemann-Stieltjes integral).  itr   Georg Friedrich Bernhard Riemann 1826 - 1866 Table of Contents
113 SOLO Vector Integration Vector Analysis Surface Integrals  vur ,  rd udru  vdrv  sd P O vdrv  udru  vdrudrvdrudrsd vuvu   Let be continuous at all points on a surface S of a finite area A, defined by the position vector . vur ,   zyxA ,,  A surface integral over the vector field is defined asA             S vu S sdnsd S vdudrrvuxvuyvuxAsdnAsdA  ,,,,,ˆ ˆ Table of Contents
114 SOLO Vector Integration Vector Analysis Volume Integrals Let be continuous at all points on a finite volume V, defined by the position vector . 321 ,, uuur    321 ,,,, uuuAzyxA   A volume integral over the vector field is defined asA      V uuu V udududrrruuuAdvA 321321 321 ,,,,  where                                                     321 333 222 111 ,, ,, det,, 321321 uuu zyx J u z u y u x u z u y u x u z u y u x rrrrrr uuuuuu   dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 Table of Contents
115 SOLO Simply and Multiply Connected Regions A region R is called simply-connected if any simple closed curve Γ, which lies in R can be shrunk to a point without leaving R. A region R that is not simply-connected is called multiply-connected. C0 x y R C1  C0 x y R C1 C2 C3  C x y R  C x y R simply-connected multiply-connected. Vector Integration Vector Analysis Table of Contents
116 SOLO Green’s Theorem in the Plane C R Let P (x,y) and Q (x,y) be continuous and have continuous partial derivatives in a region R and on the boundary C. Green’s Theorem states that:                R dydx y P x Q dyQdxP C Vector Integration Vector Analysis http://en.wikipedia.org/wiki/George_Green This Theorem was first published by George Green (1793 – 1841) in 1828 in a paper “An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism”.
117 SOLO Green’s Theorem in the Plane Vector Integration Vector Analysis Lord Kelvin rediscovered his work four years after his death and gave it wide publicity. Kelvin, James Clerk Maxwell, George Gabriel Stokes and others built on his pioneering work and Green gained a posthumous reputation amongst 19th- and 20th-century mathematicians and scientists. His work has had great influence and nowadays he is remembered principally for Green’s theorem in vector analysis, Green’s tensor (or the Cauchy-Green tensor) in elasticity theory and above all for Green’s functions for solving differential equations. George Green (1793-1841) was one of the most remarkable of nineteenth century physicists, a self-taught mathematician whose work has contributed greatly to modern physics. He was a pioneer in the application of mathematics to physical problems. He had very little formal education and died without achieving any recognition among other mathematicians. http://www.historyoftheuniverse.com/george_green/store.htm#Vol_1_paper George Green 1793-1841 tomb stone
118 SOLO Proof of Green’s Theorem in the Plane C R P T S Q a b x y  xgy 2  xgy 1 Start with a region R and the boundary curve C, defined by S,Q,P,T, where QP and TS are parallel with y axis.              b a xgy Xgy dy y P dxdydx y P 2 R By the fundamental lemma of integral calculus:                  xgxPxgxPyxPdy y yxP xgy xgy xgy Xgy 12 ,,, , 2 1 2         Therefore:          b a b a dxxgxPdxxgxPdydx y P 12 ,, R but:        a bSQ dxxgxPdxxgxP 22 ,, integral along curve SQ        b aPT dxxgxPdxxgxP 11 ,, integral along curve PT If we add to those integrals:     00,,   dxsincedxyxPdxyxP QPTS we obtain:                CTSPTQPSQ dxyxPdxyxPdxxgxPdxyxPdxxgxPdydx y P ,,,,, 12 R Assume that PT is defined by the function y = g1 (x) and SQ is defined by the function y = g2 (x), both smooth and y P   is continuous in R: Vector Integration Vector Analysis
119 SOLO Proof of Green’s Theorem in the Plane (continue – 1)      C dxyxPdydx y P , R In the same way:      C dyyxQdydx x Q , R Therefore we obtain:                R dydx y P x Q dyQdxP C The line integral is evaluated by traveling C counterclockwise. For a general single connected region, as that described in Figure to the right, can be divided in a finite number of sub-regions Ri, each of each are of the type described in the Figure above. Since the adjacent regions boundaries are traveled in opposite directions, there sum is zero, and we obtain again:                R dydx y P x Q dyQdxP C C R4 x y R R3 R1 R2 C R P T S Q a b x y  xgy 2  xgy 1 Vector Integration Vector Analysis
120 SOLO Proof of Green’s Theorem in the Plane (continue – 2) The general multiply-connected regions can be transformed in a simply connected region by infinitesimal slits C0 x y R C1 P0 P1 C0 x y R C1 C2 C3                   R dydx y P x Q dyQdxPdyQdxP i CC i0 All line integrals are evaluated by traveling Ci i=0,1,… counterclockwise. Since the slits boundaries are traveled in opposite directions, there integral sum is zero:     0 0 1 1 0   P P P P dyQdxPdyQdxP We obtain: Vector Integration Vector Analysis Table of Contents
121 SOLO Stoke’s Theorem C R Let P (x,y) and Q (x,y) be continuous and have continuous partial derivatives in a region R and on the boundary C. Green’s Theorem states that: GEORGE STOCKES 1819-1903 A more general theorem was given by Stokes                R dydx y P x Q dyQdxP C                                      yzxzxy RRR dzdy z Q y R dzdx x R z P dydx y P x Q dzRdyQdxP C or in vector form:   S dAFdrF C  where:         zzyxRyzyxQxzyxPzyxF 1,,1,,1,,,,   zdzydyxdxdr 111  zdydxydzdxxdzdydA 111  GEORGE GREEN 1793-1841 z z y y x x 111          Vector Integration Vector Analysis
122 Proof of Stoke’s Theorem SOLO GEORGE STOCKES 1819-1903 Vector Analysis A B C D v u duu  dvv  Constant v curves Constant u curves  vur ,   vuA ,  C Consider a surface in a 3 dimensional space, defined by two parameters u and v and bounded by a curve Γ. Let choose four points on this surface:  vurA ,:  vduurB ,:   dvvurD ,:  dvvduurC  ,: Consider also a vector, function of the position:  vuA ,  The vector at the four points, A, B, C, D, is given by:    vuAAA ,                  vuAdu u r vuAvuArdvuAAdvuAvduuABA uu ,,,,,,                           vuAdv v r vuAvuArdvuAAdvuAdvvuADA vv ,,,,,,                       vuAdv v r vuAdu u r vuAAdAdvuAdvvduuACA vu ,,,,,                     where du, dv are infinitesimals (differentials)
123 Proof of Stoke’s Theorem (continue – 1) SOLO Vector Analysis    vuAAA ,        vuAdu u r vuABA ,,                 vuAdv v r vuADA ,,                   vuAdv v r vuAdu u r vuACA ,,,                     A B C D v u duu  dvv  Constant v curves Constant u curves  vur ,   vuA ,  C Let compute the path integral:                 dvdu u r Adv v r v r A u r dv v r Adv v r Adu u r Adv v r Adu u r A dv v r Adv v r Adu u r Adu u r Adu u r A dr AADA dr DACA dr CABA dr BAAA drA DACDBCAB ABCD                                                                                                                                                                       2 1 2 1 2 1 2 1 2222
124 Proof of Stoke’s Theorem (continue – 2) SOLO Vector Analysis A B C D v u duu  dvv  Constant v curves Constant u curves C  vur ,   vuA ,  Let compute: dvdu u r A v r v r A u r drA ABCD                                                        u r A v r v r A u r v r u r AA u r v r u r A v r u r A A                                                                                                 Therefore:     sdAdvdu v r u r AdrA ABCD              We identify as the vector describing the surface ABCD since it is normal to surface and the aria is equal to dvdu v r u r sd               : dvdu v r u r             Let sum over the entire u,v network. Interior line integrals will cancel out in pairs leaving only , and finally we obtain:   C drA     SC sdAdrA  Table of Contents
125 Divergence Theorem This therem is also known as Gauss’ Theorem, Ostrogradsky’s Theorem or Gauss-Ostrogradsky Theorem. V A   ds SOLO Vector Analysis JOSEPH-LOUIS LAGRANGE 1736-1813 The theorem was first discovered by Lagrange in 1762, than later rediscovered by Carl Friedrich Gauss in 1813,by George Green in 1825, and in 1831 by Michail Vasilievich Ostrogradsky who gave the first proof. MIKHAIL VASILIEVICH OSTROGRADSKI 1801-1862 GEORGE GREEN 1793-1841 tomb stone http://en.wikipedia.org/Divergence_theorem JOHANN CARL FRIEDRICH GAUSS 1777-1855    S V dvAsdA 
126 Proof of Divergence Theorem SOLO Vector Analysis    S V dvAsdA         2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u     2 , 2 , 2 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu dV is any volume, that includes the point (u1,u2,u3) and is closed by the surface S=S1+S2+S3+S4+S5+S6   654321 SSSSSSS AdsAdsAdsAdsAdsAdsAds  where 32 32 3 3 2 2 41 udud u r u r ud u r ud u r dsds                      31 13 1 1 3 3 52 udud u r u r ud u r ud u r dsds                      21 21 2 2 1 1 63 udud u r u r ud u r ud u r dsds                        321 321 3 3 2 2 1 1 321 1 1 321 2 ,, 2 ,, 41 ududud u r u r u A ud u r ud u rud u A uuuA ud u A uuuAAdsAds SS                                                                      321 132 1 1 3 3 2 2 321 2 2 321 2 ,, 2 ,, 52 ududud u r u r u A ud u r ud u rud u A uuuA ud u A uuuAAdsAds SS                                                                      321 213 2 2 1 1 3 3 321 3 3 321 2 ,, 2 ,, 63 ududud u r u r u A ud u r ud u rud u A uuuA ud u A uuuAAdsAds SS                                                             
127 SOLO Vectors & Tensors in a 3D Space        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu   654321 SSSSSSS AdsAdsAdsAdsAdsAdsAds  321 213132321 ududud u r u r u A u r u r u A u r u r u A                                                          321 321 ududud u r u r u r vd                   Proof of Divergence Theorem (continue – 1)      321 21 321 13 321 32 ,, , ,, , ,, 321 uuu uu uuu uu uuu uu rrr rr u rrr rr u rrr rr u             Using the relations:                   VS vd u A u u A u u A uAds 3 3 2 2 1 1   We obtain: We also obtained the definition of Nabla ( ): 3 3 2 2 1 1 : u u u u u u          from which:   VS vdAAds  q.e.d. Table of Contents
128 VECTOR NOTATION CARTESIAN TENSOR NOTATION Gauss’ Theorem Variations   A analytic inV       A C C const vector .     S V dvsdGAUSS   2  analytic inV   S V k k dv s ds         S V dvAsdAGAUSS  1   S V k k kk dv x A dsA   SOLO Vector Analysis V A   ds Karl Friederich Gauss 1777-1855
129 VECTOR NOTATION CARTESIAN TENSOR NOTATION         S V dvAsdAGAUSS  3    V dvAA    ,A  analytic inV     S V k k kk dv x A dsA              V k k k k dv x A x A              B e e e  1 1 2 2 3 3          S V dvABBAsdABGAUSS  4            S V k k i k i kkki dv x A B x B AdsAB        A analytic inV     S V dvAAsdGAUSS  5             S V j i i j ijji dv x A x A AdsAds     SOLO Vector Analysis Table of Contents Gauss’ Theorem Variations (continue)
130 CS A   ds  dr VECTOR NOTATION CARTESIAN TENSOR NOTATION Stokes’ Theorem Variations   SC sdArdAStokes  1    A analytic on S           S k j i i j C ii sd x A x A rdA     GEORGE STOCKES 1819-1903 SOLO Vector Analysis      SSC sdAAsdArdAStokes  2   SC sdrdStokes   3     AsdsdAA const    
131 SOLO Vector Analysis vectorconstCCAA .       SC sdCArdCA             CsdAnCsdAnnAC sdACsdACArdC SS sdnsd SSC         ˆˆˆ ˆ     A d r A d s C S          SC sdAnArdStokes  ˆ4      ACACCA constC         ArdCrdCA              AnnAAn AnnAAn A A       ˆˆˆ ˆˆˆ        AnAnAnAn   ˆˆˆˆ           SSC sdAnAnAnsdAnArdStokes  ˆˆˆˆ4 Stokes’ Theorem Variations (continue - 1)
132 GAUSS’AND STOKES’ THEOREMS ARE GENERALIZATIONS OF THE FUNDAMENTAL THEOREM OF CALCULUS  A b A a d A x d x d x a b ( ) ( )   SOLO Vector Analysis    SC sdAnArdStokes  ˆ4 Use with rA         nnnrnrnrn r ˆ2ˆ3ˆˆˆˆ 3     therefore   SC sdnrdrStokes ˆ 2 1 5  Stokes’ Theorem Variations (continue - 2) Table of Contents
133 SOLO GREEN’s IDENTITIES Start from Gauss’ Theorem that relates the integral of the flux of a union of closed surfaces to it’s divergence. n i iSS 1          S GAUSS V dSnFGdvFG 1   n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   and must be continuous and twice differentiable in V.G  F  Using the identity we obtain         FGFGFG                S GAUSS V dSnFGdvFGFG 1  First Vector Green Identity Interchanging and we obtainG  F               S GAUSS V dSnGFdvGFFG 1  By subtracting the second identity from the first we obtain               SV dSnGFFGdvFGGF 1  Second Vector Green Identity Vector Analysis GEORGE GREEN 1793-1841 Table of Contents Harmonic
134 SOLO Derivation of Nabla (  ) from Gauss’ Theorem Start from Gauss’ Theorem that relates the integral of the flux of a union of closed surfaces to it’s divergence. n i iSS 1   n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Vector Analysis                             ,, 1 lim: 1 lim, 1 lim, 1 lim, 0 0 0 0 S V S V F S VF S V F ds V Ads V trA Ads V trA ds V tr                             S V F S VF S V F Ads V trA Ads V trA ds V tr    1 lim, 1 lim, 1 lim, 0 0 0                 S GAUSS V S GAUSS V S GAUSS V AdsvdA AdsvdA dsvd   5 1 2 Table of Contents
135 SOLO Vectors & Tensors in a 3D Space The Operator .                             ,, 1 lim: 1 lim, 1 lim, 1 lim, 0 0 0 0 S V S V F S VF S V F ds V Ads V trA Ads V trA ds V tr   We know that where V is any volume, that includes the point and is closed by the surface SFr         2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu Let apply those definitions to the infinitesimal volume in the figure, having the point at it’s centerFr  where 321 321 3 3 2 2 1 1 ududud u r u r u r ud u r ud u r ud u r V                                        S VF ds V tr 1 lim, 0  Gradient V is any volume, that includes the point (u1,u2,u3) and is closed by the surface S=S1+S2+S3+S4+S5+S6
136 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 1) where        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu   654321 SSSSSSS dsdsdsdsdsdsds 32 32 3 3 2 2 41 udud u r u r ud u r ud u r dsds                      31 13 1 1 3 3 52 udud u r u r ud u r ud u r dsds                      21 21 2 2 1 1 63 udud u r u r ud u r ud u r dsds                        321 321 3 3 2 2 1 1 321 1 1 321 2 ,, 2 ,, 41 ududud u r u r u ud u r ud u rud u uuu ud u uuudsds SS                                                              321 132 1 1 3 3 2 2 321 2 2 321 2 ,, 2 ,, 52 ududud u r u r u ud u r ud u rud u uuu ud u uuudsds SS                                                              321 213 2 2 1 1 3 3 321 3 3 321 2 ,, 2 ,, 63 ududud u r u r u ud u r ud u rud u uuu ud u uuudsds SS                                                     
137 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 2) or        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu   654321 SSSSSSS dsdsdsdsdsdsds 321 213132321 ududud u r u r uu r u r uu r u r u                                                      321 321 ududud u r u r u r V                                                                            321 321213132 0 1 lim u r u r u r uu r u r uu r u r uu r u r ds V S V   3 3 2 2 1 1 u u u u u u          To the same result we could arrive using: rdu u u u u u rdu u rdu u rdu u ud u ud u ud u rdd                                      3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 Gradient
138 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 3) or        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu                                                          321 321213132 0 1 lim u r u r u r u A u r u r u A u r u r u A u r u r Ads V A S V    3 3 2 2 1 1 u A u u A u u A uA            By the same procedure as before: Divergence    S V Ads V A  1 lim 0 or                                                          321 321213132 0 1 lim u r u r u r u A u r u r u A u r u r u A u r u r Ads V A S V    3 3 2 2 1 1 u A u u A u u A uA            By the same procedure as before: Rotor    S V Ads V A  1 lim 0 Summarize 3 3 2 2 1 1 u u u u u u         
139 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 4) Let develop those equations in curvilinear coordinates: Therefore:   i i u       j ij j iji u gg    We obtain:   ii u i ij u i ij i i r u ggr u gu u uuu 1,, 321           where: g r r r r i i i i u u u u    1 Second Proof:           j j jij i j ji i ud u udgud u r u r u r u r ud u r ud u r ud u r rdd                                        3 32 1 1 3 3 2 2 1 1   j iji u g     ij i j g u      i i u u u u u u u uuuu             3 3 2 2 1 1321 ,,Gradient
140 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 5) Let develop those equations in curvilinear coordinates: j ju i uArAA i   Divergence     i ij ju i u A uuArAA i      But j m m ij i j u mj mi i j i uA u A rA u A u A j                            Therefore   ij ij j g ji m m ji i j u imj im i j uuA u A ruA u A A                         or ij m m ji i jmi im i i gA u A A u A A                        
141 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 6) Divergence (continue – 1)j ju i uArAA i   ij m m ij i jmi im i i gA u A A u A A                         Let compute i im konsummationggGggG ki iki jkj ik   ik ik G g g    m kiki m kiik m ki kim u g gg u g G u g g g u g                                                  k im i mk m ik ik mi i im mj k ij i jk j ik km kij km ij m u g u g u gg u g u g u gg g 22 ,Start from But k miki gg k imki ki i mkik u g g u g g u g g miim          m ik ki mi i u gg    2 We found mmm ik ki mi i u g gu g gu gg          1 2 1 2   i i i ii iim m mi i mi im i i u Ag g A u g gu A A u g gu A A u A A                    111   ij m m ij i jmi im i i i i gA u A A u A u Ag g A                           1
142 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 5) Rotor     i ij ju i u A uuArAA i      But j m m ij i j u mj mi i j i uA u A rA u A u A j                            Therefore ji m m ji i j u imj im i j uuA u A ruA u A A j                         j ju i uArAA i   Let develop those equations in curvilinear coordinates:       otherwise ofnpermutatiocyclicakji ofnpermutatiocyclicakji r g uu kjiu kjiji k 0 3,1,2,,1 3,2,1,,1 ,, ,,    1,2,3ofnspermutatiocyclicarekj,i, g r u A u A g r A u A A u A A kji u j i i j kji u m m ij j i m m ji i j k k                                                ,, ,,    Use the fact that and are reciprocal vectors we have i uju r 
143 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 6) Laplacian Δ j ju i uArAA i   Let develop those equations in curvilinear coordinates:    ij m m ij ji j ji ii i g uuu u gg ugu g g                               2 2 11   i i u       j ij j iji u gg    Using   ij m m ij i jmi im i i i i gA u A A u A u Ag g A                           1   ii u i ij u i ij i i r u ggr u gu u uuu 1,, 321           We found Table of Contents
144 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space Let express the cartesiuan coordinates (x, y, z) of any point, in a three dimensional space as a function of three curvilinear coordinates (u1, u2, u3), where:  dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3                 zyxuuzyxuuzyxuu uuuzuuuyuuuxx ,,,,,,,, ,,,,,,,, 332211 321321321 Those functions are single valued with continuous derivatives and the correspondence between (x,y,z) and (u1,u2,u3) is unique (isomorphism). kzjyixr   kzdjydixdrd   For orthogonal coordinates we have: 3332221113 3 2 2 1 1 eudheudheudhud u r ud u r ud u r rd                    ji ji ee ji 0 1      2 3 2 3 2 2 2 2 2 1 2 1 2 udhudhudhrdrdsd   3213213 3 2 2 1 1 udududhhhud u r ud u r ud u r Vd                   33 3 22 2 11 1 /:/:/: u r u r e u r u r e u r u r e                     
145 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space (continue – 1)  dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3 General Coordinates:       3 3 321 22113 2 2 321 11332 1 1 321 33221 321 21 321 13 321 32 ,, , ,, , ,, h e hhh eheh rrr rr u h e hhh eheh rrr rr u h e hhh eheh rrr rr u uuu uu uuu uu uuu uu                     33 3 22 2 11 1 321 ,, eh u r reh u r reh u r r uuu                  i i u u     Orthogonal Coordinates: i ii e uh     1 Gradient: j ju i uri  AAA    321321 ,, hhhrrrg uuu   33 3 22 2 11 1 uh e uh e uh e          
146 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space (continue – 2)  dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3 General Coordinates:       3 3 321 22113 2 2 321 11332 1 1 321 33221 321 21 321 13 321 32 ,, , ,, , ,, h e hhh eheh rrr rr u h e hhh eheh rrr rr u h e hhh eheh rrr rr u uuu uu uuu uu uuu uu                     33 3 22 2 11 1 321 ,, eh u r reh u r reh u r r uuu                  Orthogonal Coordinates:    321 33221133 3 3 22 2 2 11 1 1 3 33 2 22 1 11 3 3 33 2 2 22 1 1 11332211 /// uuu rhArhArhAeh h A eh h A eh h A uhAuhAuhA h e hA h e hA h e hAeAeAeA          321 321 AAA AAA A     i i u Ag g    1 ADivergence:                      3 3 21 2 2 31 1 1 32 321 1 u Ahh u Ahh u Ahh hhh A j ju i uri  AAA    321321 ,, hhhrrrg uuu  
147 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space (continue – 3) General Coordinates:       3 3 321 22113 2 2 321 11332 1 1 321 33221 321 21 321 13 321 32 ,, , ,, , ,, h e hhh eheh rrr rr u h e hhh eheh rrr rr u h e hhh eheh rrr rr u uuu uu uuu uu uuu uu                     33 3 22 2 11 1 321 ,, eh u r reh u r reh u r r uuu                  Orthogonal Coordinates:Rotor: 1,2,3ofnspermutatiocyclicarekj,i, g r u A u A kji u j i i j k                ,,  A j ju i uri  AAA     321 33221133 3 3 22 2 2 11 1 1 3 33 2 22 1 11 3 3 33 2 2 22 1 1 11332211 /// uuu rhArhArhAeh h A eh h A eh h A uhAuhAuhA h e hA h e hA h e hAeAeAeA          321 321 AAA AAA A     321321 ,, hhhrrrg uuu   332211 321 332211 AhAhAh uuu eheheh         A  dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3
148 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space (continue – 4) General Coordinates:       3 3 321 22113 2 2 321 11332 1 1 321 33221 321 21 321 13 321 32 ,, , ,, , ,, h e hhh eheh rrr rr u h e hhh eheh rrr rr u h e hhh eheh rrr rr u uuu uu uuu uu uuu uu                     33 3 22 2 11 1 321 ,, eh u r reh u r reh u r r uuu                  Orthogonal Coordinates: Laplacian: j ju i uri  AAA    321321 ,, hhhrrrg uuu    dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3              j ji i u gg ug 1       ji jih h e h e uug i j j i ijiji 0 /1 2                                          33 21 322 31 211 32 1321 1 uh hh uuh hh uuh hh uhhh Table of Contents
149 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems       z y x z y x          1. Gradient  • Cartesian:       zyx zyx 111         z r r z r            1 • Cylindrical:       zr zr 111                       sin 1 1 r r r r • Spherical:         111   rr
150 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 2. Divergence  • Cartesian: z A y A x A A zyx           • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      z AA r Ar rr A z r           11   111   AAAA rr                    A r A r Ar rr A r sin 1 sin sin 11 2 2 
151 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 3. Laplacian 2  • Cartesian: 2 2 2 2 2 2 2 zyx          • Cylindrical: • Spherical: 2 2 2 2 22 2 2 2 2 2 2 2 1111 zrrrrzrr r rr                                      2 2 222 2 2 2 sin 1 sin sin 11                        rrr r rr
152 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 4. Curl  • Cartesian:       y A x A A x A z A A z A y A A xy z zx y yz x                      • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      111   AAAA rr        zyx zyx AAAA 111                                         r z zr z r A Ar rr A r A z A A z AA r A 1 1          zr zr AAAA 111                                                        r r r A Ar rr A Ar r A r A A A r A 1 sin 1 sin sin 1            111   AAAA rr 
153 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 5. Scalar Product • Cartesian: • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      111   AAAA rr  zyx zyx BBBB 111    zzyyxx BABABABA   zr zr BBBB 111       111   BBBB rr  zzrr BABABABA     BABABABA rr  
154 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 6. Vector Product • Cartesian: • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      111   AAAA rr  zyx zyx BBBB 111          zyx zyx BABABABA 111    zr zr AAAA 111       111   AAAA rr        xyyxz zxxzy yzzyx BABABA BABABA BABABA             zr zr BABABABA 111           rrz zrrz zzr BABABA BABABA BABABA                rr rr r BABABA BABABA BABABA                  111   BABABABA rr 
155 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 7. Material Derivative • Cartesian: • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      111   AAAA rr  zyx zyx vvvv 111    z A v y A v x A v t A tD AD z A v y A v x A v t A tD AD z A v y A v x A v t A tD AD z z z y z x z z y z y y y x y y x z x y x x x x                                                               Av t A tD AD       zr zr vvvv 111       111   vvvv rr  z A v A r v r A v t A tD AD z A v A r v r A v t A tD AD z A v A r v r A v t A tD AD z z zz r z z zr r z rr r r r                                                                                                                                     A r vA r v r A v t A tD AD A r vA r v r A v t A tD AD A r vA r v r A v t A tD AD r r rrr r r r sin sin sin    Table of Contents
156 Vector AnalysisSOLO Applications Fundamental Theorem of Vector Analysis for a Bounded Region V (Helmholtz’s Theorem) Reynolds’ Transport Theorem Poisson’s Non-homogeneous Differential Equation Kirchhoff’s Solution of the Scalar Helmholtz Non-homogeneous Differential Equation Table of Contents Fundamental Theorem of Vector Analysis for a Unbounded Region V (Helmholtz’s Theorem) Laplace Fields Harmonic Functions Rotations
157 ROTATIONS Rotation of a Rigid Body SOLO 23r 31r 12r1 3 2 P P 1 2 331r 23r 12r A rigid body in mechanics is defined as a system of mass points subject to the constraint that the distance between all pair of points remains constant through the motion. To define a point P in a rigid body it is enough to specify the distance of this point to three non-collinear points. This means that a rigid body is completely defined by three of its non-collinear points. Since each point, in a three dimensional space is defined by three coordinates, those three points are defined by 9 coordinates. But the three points are constrained by the three distances between them: 313123231212 && constrconstrconstr  Therefore a rigid body is completely defined by 9 – 3 = 6 degrees of freedom. This is a part of the Presentation “ROTATIONS” NOTES ON ROTATIONS SOLO HERMELIN INITIAL INTERMEDIATE FINAL
158 ROTATIONS Rotation of a Rigid Body (continue – 1) SOLO We have the following theorems about a rigid body: Euler’s Theorem (1775) The most general displacement of a rigid body with one point fixed is equivalent to a single rotation about some axis through that point. Chasles’ Theorem (1839) The most general displacement of a rigid body is a translation plus a rotation. Leonhard Euler 1707-1783 Michel Chasles 1793-1880
159 ROTATIONS Rotation of a Rigid Body (continue – 2) SOLO Proof of Euler’s Theorem P 'P OA 'A B 'BC C r rr r r   O – Fixed point in the rigid body A,B – Two point in the rigid body at equal distance r from O.        rOBOA __________ A’,B’ – The new position of A,B respectively. Since the body is rigid rOBOA  __________ '' Therefore A,B, A’,B’ are one a sphere with center O.  – plane passing through O such that A and A’ are at the same distance from it.  – plane passing through O such that B and B’ are at the same distance from it. PP’ – Intersection of the planes and  The two spherical triangles APB and A’PB’ are equal. The arcs AA’ and BB’ are equal. That means that rotation around PP’ that moves A to A’ will move B to B’. q.e.d.
160 ROTATIONS Mathematical Computation of a Rotation SOLO A B C O   nˆ v  1v  We saw that every rotation is defined by three parameters: • Direction of the rotation axis , defined by by two parameters.nˆ • The angle of rotation , defines the third parameter. Let rotate the vector around by a large angle , to obtain the new vector   OAv  nˆ   OBv1  From the drawing we have:   CBACOAOBv1  vOA      cos1ˆˆ   vnnAC  Since direction of is:      sinˆˆ&ˆˆ  vnnvnn  and it’s length is: AC    cos1sin v   sinˆ vnCB    Since has the direction and the absolute value CB  vn  ˆ sinsinv       sinˆcos1ˆˆ1 vnvnnvv  
161 ROTATIONS Computation of the Rotation Matrix SOLO We have two frames of coordinates A and B defined by the orthogonal unit vectors and AAA zyx ˆ,ˆ,ˆ  BBB zyx ˆ,ˆ,ˆ The frame B can be reached by rotating the A frame around some direction by an angle .nˆ  We want to find the Rotation Matrix that describes this rotation from A to B.  ,ˆ33 nRC x B A                    sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ AAAB AAAB AAAB znznnxz ynynnxy xnxnnxx    Let write those equations in matrix form.                                               0 0 1 sinˆ 0 0 1 cos1ˆˆ 0 0 1 ˆ  AAAA B nnnx                  0 0 0 ˆ xy xz yz A nn nn nn n   0ˆ ntrace Axˆ Azˆ Ayˆ Bzˆ Byˆ Bxˆ O nˆ     Rotation Matrix
162 ROTATIONS Computation of the Rotation Matrix (continue – 1) SOLO Axˆ Azˆ Ayˆ Bzˆ Byˆ Bxˆ O nˆ                                                   0 0 1 sinˆ 0 0 1 cos1ˆˆ 0 0 1 ˆ  AAAA B nnnx                                               0 1 0 sinˆ 0 1 0 cos1ˆˆ 0 1 0 ˆ  AAAA B nnny                                               1 0 0 sinˆ 1 0 0 cos1ˆˆ 1 0 0 ˆ  AAAA B nnnz                   A A A B AAA x A B xCnnnIx ˆ 0 0 1 sinˆcos1ˆˆˆ 33                                A A A B AAA x A B yCnnnIy ˆ 0 1 0 sinˆcos1ˆˆˆ 33                                A A A B AAA x A B zCnnnIz ˆ 1 0 0 sinˆcos1ˆˆˆ 33              Rotation Matrix (continue – 1)
163 ROTATIONS Computation of the Rotation Matrix (continue – 2) SOLO Ax Az Ay Bz By Bx  O nˆ                      ,ˆsinˆcos1ˆˆ 3333 nRnnnICC x AAA x A B A B  The matrix has the following properties:  A nˆ        ATA nn  ˆˆ                                                   22 22 22 0 0 0 0 0 0 ˆˆ yxzyzx zyzxyx zxyxyz xy xz yz xy xz yz AA nnnnnn nnnnnn nnnnnn nn nn nn nn nn nn nn T x zzyzx zyyyx zxyxx nnI nnnnn nnnnn nnnnn ˆˆ 000 010 001 33 2 2 2                                  213ˆˆ  AA nntrace        nn nn nn nn nnnnn xy xz yz zyx AT ˆˆ000 0 0 0 ˆˆ                                     AATAAT x AAA nnnnnnnnInnn  ˆˆˆˆˆˆˆˆˆˆˆ 22                    T x AAAAAA nnInnnnnn ˆˆˆˆˆˆˆˆ 33  skew-symmetric Rotation Matrix (continue – 2)
164 ROTATIONS Computation of the Rotation Matrix (continue – 3) SOLO Ax Az Ay Bz By Bx  O nˆ                                              B Axx AAA x TATATA x TA B CnRnR nnnI nnnIC       ,ˆ,ˆ sinˆcos1ˆˆ sinˆcos1ˆˆ 3333 33 33 Note The last term can be writen in matrix form as Therefore In the same way End Note In fact is the matrix representation of the vector product:  vnn   ˆˆ    vInnvvnn x T  33 ˆˆˆˆ          vvnnnnvvnnvnn   ˆˆˆˆˆˆˆˆ    T x nnInn ˆˆˆˆ 33              nnnnvnvvnnnvnnn ˆˆˆˆˆˆˆˆˆˆˆ                nnnnnnvnnvnnnn ˆˆˆˆˆˆˆˆˆˆˆˆ  Rotation Matrix (continue – 3)
165 ROTATIONS Computation of the Rotation Matrix (continue – 4) SOLO Ax Az Ay Bz By Bx  O nˆ                                            sin0cos123 sinˆcos1ˆˆ sinˆcos1ˆˆ 33 33    AAA x TATATA x B A ntracenntraceItrace nnnItracetraceC Therefore cos21 B ACtrace Let compute the trace (sum of the diagonal components of a matrix) of B AC Also we have                                              sinˆcos1ˆˆcos sinˆcos1ˆˆ sinˆcos1ˆˆ 33 3333 33 AT x AT xx TATATA x B A nnnI nnnII nnnIC    sin 0 0 0 cos1cos 000 010 001 2 2 2                                       xy xz yz zzyzx zyyyx zxyxx nn nn nn nnnnn nnnnn nnnnn Rotation Matrix (continue – 4)
166 ROTATIONS Computation of the Rotation Matrix (continue – 5) SOLO Ax Az Ay Bz By Bx  O nˆ     Therefore we have                                      cos1cossincos1sincos1 sincos1cos1cossincos1 sincos1sincos1cos1cos 2 2 2 zxzyyzx xzyyzyx yzxzyxx B A nnnnnnn nnnnnnn nnnnnnn C We get  1 2 1 cos  B AtraceC two solutions for  If ; i.e. we obtain0sin   ,0       sin2/2,33,2 B A B Ax CCn        sin2/3,11,3 B A B Ay CCn        sin2/1,22,1 B A B Az CCn  Rotation Matrix (continue – 5)
167 ROTATIONS Consecutive Rotations SOLO - Perform first a rotation of the vector , according to the Rotation Matrix to the vector . v   1133 ,ˆ nR x 1v  - Perform a second a rotation of the vector , according to the Rotation Matrix to the vector . 1v   2233 ,ˆ nR x 2v   vnRv x  11331 ,ˆ         vnRvnRnRvnRv xxxx   ,ˆ,ˆ,ˆ,ˆ 3311332233122332  The result of those two consecutive rotation is a rotation defined as:      1133223333 ,ˆ,ˆ,ˆ  nRnRnR xxx  Let interchange the order of rotations, first according to the Rotation Matrix and after that according to the Rotation Matrix .  2233 ,ˆ nR x  1133 ,ˆ nR x The result of those two consecutive rotation is a rotation defined as:    22331133 ,ˆ,ˆ  nRnR xx Since in general, the matrix product is not commutative        2233113311332233 ,ˆ,ˆ,ˆ,ˆ  nRnRnRnR xxxx  Therefore, in general, the consecutive rotations are not commutative. Rotation Matrix (continue – 6)
168 ROTATIONSSOLO INITIALINITIAL INTERMEDIATEINITIAL INTERMEDIATE FINAL Consecutive Rotations of a DiceRotation Matrix (continue – 7)
169 ROTATIONS Decomposition of a Vector in Two Different Frames of Coordinates SOLO We have two frames of coordinate systems A and B, with the same origin O. We can reach B from A by performing a rotation. Let describe the vector in both frames.v  Axˆ Azˆ Ayˆ Bxˆ Bzˆ Byˆ v  O xAv zAv yAv xBv zBv yBv BzBByBBxBAzAAyAAxA zvyvxvzvyvxvv  111111               zA yA xA A v v v v               zB yB xB B v v v v  &                   BBABBABBAA BBABBABBAA BBABBABBAA zzzyyzxxzz zzyyyyxxyy zzxyyxxxxx ˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ1ˆˆ                          zABBABBABBA yABBABBABBA xABBABBABBA vzzzyyzxxz vzzyyyyxxy vzzxyyxxxxv ˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ1ˆ     from which Rotation Matrix (continue – 8)
170 ROTATIONS Decomposition of a Vector in Two Different Frames of Coordinates (continue – 1) SOLO                                                     zA yA xA BABABA BABABA BABABA zB yB xB v v v zzzyzx yzyyyx xzxyxx v v v ˆˆˆˆˆˆ ˆˆˆˆˆˆ ˆˆˆˆˆˆ Axˆ Azˆ Ayˆ Bxˆ Bzˆ Byˆ v  O xAv zAv yAv xBv zBv yBv   AB A B vCv   where is the Transformation Matrix (or Direction Cosine Matrix – DCM) from frame A to frame B. B AC                                BABABA BABABA BABABA B A B A zzzyzx yzyyyx xzxyxx CC ˆˆˆˆˆˆ ˆˆˆˆˆˆ ˆˆˆˆˆˆ : In the same way        BA B BB A A vCvCv   1 therefore  1  B A A B CC Rotation Matrix (continue – 9)
171 ROTATIONS Decomposition of a Vector in Two Different Frames of Coordinates (continue – 2) SOLO Axˆ Azˆ Ayˆ Bxˆ Bzˆ Byˆ v  O xAv zAv yAv xBv zBv yBv                    ATAAB A TB A TAAB A TAB A BTB vvvCCvvCvCvvv  2 Since the scalar product is independent of the frame of coordinates, we have      1  B A TB A B A TB A CCICC                                                                           100 010 001 3,33,23,1 2,32,22,1 1,31,21,1 3,32,31,3 3,22,21,2 3,12,11,1 B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A TB A CCC CCC CCC CCC CCC CCC CC or            3,2,10 3,2,11 ,, 3 1 jji iji kjCkiC ij k B A B A  Those are 9 equations in , but by interchanging i with j we get the same conditions, therefore we have only 6 independent equations.   3,2,1,, jijiC B A We see that the Rotation Matrix is ortho-normal (having real coefficients and the rows/columns are orthogonal to each other and of unit absolute value. Rotation Matrix (continue – 10) This means that the relation between the two coordinate systems is defined by 9 – 6 = 3 independent parameters.
172 ROTATIONS Differential Equations of the Rotation Matrices SOLO We want to develop the differential equation of the Rotation Matrix as a function of the Angular Velocity of the Rotation. Let define by: -the Rotation Matrix that defines a frame of coordinates B at the time t relative to some frame A.  tC B A -the Rotation Matrix that defines the frame of coordinates B at the time t+Δt relative to some frame A.  ttC B A    ,ˆ33xR -the Rotation Matrix from the frame of coordinates B at the time t to B at time t+Δt relative to some frame A.      tCRttC B Ax B A   ,ˆ33 and                                                   2 cos 2 sinˆ2 2 sinˆˆ2 sinˆcos1ˆˆ,ˆ 2 33 3333      x xx I IR Rotation Matrix (continue – 11)
173 ROTATIONS Differential Equations of the Rotation Matrices (continue – 1) SOLO Let differentiate the Rotation Matrix                      tC dt dIR tC t IR tC t IR t tCtCR t tCttC t C dt dC B A xxB A xx t B A xx t B A B Ax t B A B A t B A t B A                                                3333 0 3333 0 3333 0 33 0 00 ,ˆ lim ,ˆ lim ,ˆ lim ,ˆ lim limlim                                                                                  ˆ 2 cos 2 2 sin ˆ 2 2 2 sin ˆˆlim ,ˆ lim 2 2 0 3333 0 xx IR and Therefore      tC dt d dt tdC B A B A    ˆ Rotation Matrix (continue – 12)
174 ROTATIONS Differential Equations of the Rotation Matrices (continue – 2) SOLO The final result of the Rotation Matrix differentiation is: Since defines the unit vector of rotation and the rotation rate from B at time t to B at time t+Δt, relative to A, then is the angular velocity vector of the frame B relative to A, at the time t ˆ dt d   ˆ dt d      ˆ dt dB AB          tCt dt tdC B A B AB B A    By changing indixes A and B we obtain        tCt dt tdC A B A BA A B    Rotation Matrix (continue – 13)
175 ROTATIONS Differential Equations of the Rotation Matrices (continue – 3) SOLO Let find the relation between and  B AB     A AB   For any vector let perform the following computationsv            A AB B A B AB BB AB vCvv                   BA B A AB B A AB A A B A AB B A AA AB B A vCCvCCCvC     Since this is true for any vector we havev        A B A AB B A B AB CC     Pre-multiplying by and post-multiplying by we get: A BC B AC       B A B AB A B A AB CC     Rotation Matrix (continue – 14)
176 ROTATIONS Differential Equations of the Rotation Matrices (continue – 4) SOLO Let differentiate the equation 33x A B B A ICC  to obtain       0  dt dC C dt dC CCC dt dC CC dt dC A BB A B AB A BB A A B B A B AB A BB A A B B A   Post-multiplying by we get A BC          A B A AB A B B A B AB A B B AB A B A B CCCCC dt dC     We obtained for the differentiation of the Rotation Matrix                    B AB A B A B A AB A B A BA A B ttCtCttCt dt tdC     Note We can see that            tttt ABBA A AB A BA     End Note Rotation Matrix (continue – 15)
177 ROTATIONS Differential Equations of the Rotation Matrices (continue – 5) SOLO Suppose that we have a third frame of coordinates I (for example inertial) and we have the angular velocity vectors of frames A and B relative to I. We have    B I B IB B I C dt dC       A I A IA A I C dt dC    A I B A B I CCC  dt dC CC dt dC dt dC A IB A A I B A B I        I A A I A IA B A I A B I B IB I A A IB A I A B I B A CCCCCC dt dC CC dt dC dt dC     or From which we get:      A IA B A B A B IB B A CC dt dC     Rotation Matrix (continue – 16)
178 ROTATIONSSOLO From the equation Computation of the Angular Velocity Vector from .AB     nRtC x B A ˆ,33         tCt dt tdC B A B AB B A    we obtain         TB A B AB AB tC dt tdC t   Since the Rotation Matrix is defined also by and          sinˆcos1ˆˆcosˆ, 3333   nnnInRC T x B A  tC B A nˆ we can compute as function of and their derivativesnˆAB  td d   td nd n ˆ ˆ   (this is a long procedure described in the complementary work “Notes on Rotations”, and a simpler derivation will be given later, we give here the final result)      sinˆcos1ˆˆˆ    nnnnAB  Rotation Matrix (continue – 17)
179 ROTATIONSSOLO Computation of and as functions of .AB  td d   td nd n ˆ ˆ   Let pre-multiply the equation by and use T nˆ     sinˆcos1ˆˆˆ    nnnnAB    0ˆˆ,0ˆˆ,1ˆˆ   nnnnnn TTT to obtain     AB TTTT AB T nnnnnnnnn       ˆsinˆˆcos1ˆˆˆˆˆˆ Let pre-multiply the equation by and use nˆ     sinˆcos1ˆˆˆ    nnnnAB           nnInnnnnnn x T ˆˆˆˆˆˆˆ,0ˆˆ 33 to obtain                 sinˆˆcos1ˆsinˆˆcos1ˆˆˆˆˆˆ    nnnnnnnnnnn AB  Let pre-multiply the equation by       sinˆˆcos1ˆˆ    nnnn AB   nˆ               cos1ˆˆsinˆsinˆˆˆcos1ˆˆˆˆ    nnnnnnnnnn AB  Rotation Matrix (continue – 18)
180 ROTATIONS Computation of and as functions of (continue – 1) SOLO AB  td d   td nd n ˆ ˆ   We have two equations:       ABnnnn      ˆsinˆˆcos1ˆ        ABnnnnn      ˆˆcos1ˆˆsinˆ with two unknowns and  nˆ     nn ˆˆ From those equations we get:            sinˆˆcos1ˆsincos1ˆ 22 ABAB nnnn     or           sinˆˆcos1ˆcos1ˆ2 ABAB nnnn     Finally we obtain: AB T n    ˆ      ABnnnn                   2 cotˆˆˆ 2 1 ˆ Rotation Matrix (continue – 19)
181 ROTATIONS Quaternions SOLO The quaternions method was introduced by Hamilton in 1843. It is based on Euler Theorem (1775) that states: The most general displacement of a rigid body with one point fixed is equivalent to a single rotation about some axis through that point. Therefore every rotation is defined by three parameters: • Direction of the rotation axis , defined by two parameters • The angle of rotation , defines the third parameter nˆ  William Rowan Hamilton 1805 - 1865       sinˆcos1ˆˆ1 vnvnnvv   The rotation of around by angle is given by:nˆ v  A B C O   nˆ v  1v  that can be writen        sinˆcos1ˆˆ1 vnvvnnvv   or       sinˆcos1ˆˆcos1 vnvnnvv  
182 ROTATIONS Quaternions (continue – 1) SOLO Computation of the Rotation Matrix We found the Rotation Matrix that describes this rotation from A to B.  ,ˆ33 nRC x B A                    sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ AAAB AAAB AAAB znznnxz ynynnxy xnxnnxx    Axˆ Azˆ Ayˆ Bzˆ Byˆ Bxˆ O nˆ                       A A A B AAA x A B xCnnnIx ˆ 0 0 1 sinˆcos1ˆˆˆ 33                                A A A B AAA x A B yCnnnIy ˆ 0 1 0 sinˆcos1ˆˆˆ 33                                A A A B AAA x A B zCnnnIz ˆ 1 0 0 sinˆcos1ˆˆˆ 33              or from which                  ,ˆsinˆcos1ˆˆ 3333 nRnnnICC x AAA x A B A B 
183 ROTATIONS Quaternions (continue – 2) SOLO Definition of the Quaternions Axˆ Azˆ Ayˆ Bzˆ Byˆ Bxˆ O nˆ     The quaternions (4 parameters) were defined by Hamilton as a generalization of the complex numbers   32100 , qkqjqiqqq     2/cos0 q  nˆ2/sin          zyx nqnqnq 2/sin&2/sin&2/sin 111   where satisfy the relations:kji  ,, 1 kkjjii  kijji   , ijkkj   , jkiik   1 kji  i  j  k  the complex conjugate of is defined asq   32100 * , qkqjqiqqq   
184 ROTATIONS Quaternions (continue – 3) SOLO Product of Quaternions Product of two quaternions andAq Bq      3210321000 ,, BBBBAAAABBAABA qkqjqiqqkqjqiqqqqq         3210321033221100 AAABBBBABABABABA qkqjqiqqkqjqiqqqqqqqqq        122131132332 BABABABABABA qqqqkqqqqjqqqqi   therefore        BAABBABABABBAABA qqqqqqqq    000000 ,,, Let use this expression to find       2 3 2 2 2 1 2 0 222 000 * 00 * 1ˆˆ 2 sin 2 cos,,,, qqqqnnqqqqqqqqq                 The quaternion product can be writen in matrix form as:                                            A A BxBB T BB B B AxAA T AA BA q Iq qq Iq q qq q q           0 330 00 330 00 1 kjikkjjii  kijji   ijkkj   jkiik  
185 ROTATIONS Quaternions (continue – 4) SOLO Rotation Description Using the Quaternions Let compute the expression:                                               AAAAAAAA AAAAA vvqvqvqvvvqqv qvvqvqvqqvq 00 2 000 0000 * , ,,,,0,                                                   A AAAA AAAAAAA vqq vvqqvv vvqvqvqvvv          22,0 2,0 ,0 0 2 0 0 2 0 00 2 0 Using the relations:                                       nnq nnnn q n q ˆsinˆ2/sin2/cos22 ˆˆcos1ˆˆ2/sin22 1 ˆ2/sin 2/cos 0 2 2 0 0          and                     AAAA x AB A B vnnnIvCv   sinˆcos1ˆˆ33  we obtain                         AAABB vqqvqqvqvv    221,0,,0,,0 000 *
186 ROTATIONS Quaternions (continue – 5) SOLO Rotation Description Using the Quaternions (continue – 1) Using the fact that we obtain:          22 033 qIC x B A                                                     0 0 0 0 0 0 2 0 0 0 2 100 010 001 12 13 23 12 13 23 12 13 23 0 qq qq qq qq qq qq qq qq qq q                                          2 2 2 13231 32 2 1 2 321 3121 2 2 2 3 1020 1030 2030 2222 2222 2222 022 202 220 100 010 001 qqqqqq qqqqqq qqqqqq qqqq qqqq qqqq                 2 2 2 110323120 3210 2 1 2 33021 20312130 2 2 2 3 2212222 2222122 2222221 qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq 1 2 3 2 2 2 1 2 0  qqqq                             2 3 2 2 2 1 2 010323120 3210 2 3 2 2 2 1 2 03021 20312130 2 3 2 2 2 1 2 0 22 22 22 qqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqqq C B A
187 ROTATIONS Quaternions (continue – 6) SOLO Rotation as a Multiplication of Two Matrices          22 033 qIC x B A           22 033 2 0 qIq x T               33033 2 0 2 x T x IqIq                  33330330 x T xx IIqIq For any vector we can write         aaaa  or in matrix notation           T x T x T x TT IIaIa    333333 Therefore we have                  33330330 x T xx B A IIqIqC            T xx IqIq   330330  321 3 2 1 012 103 230 012 103 230 qqq q q q qqq qqq qqq qqq qqq qqq                                      
188 ROTATIONS Quaternions (continue – 7) SOLO Rotation as a Multiplication of Two Matrices (continue – 1)                                                                    330 330 012 103 230 321 0123 1032 2301 x T x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C                                                                      330 330 012 103 230 321 0123 1032 2301 x T x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C                                                       T x x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C             330 330 321 012 103 230 3012 2103 1230                                                         T x x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C             330 330 321 012 103 230 3012 2103 1230
189 ROTATIONS Quaternions (continue – 8) SOLO Relation Between Quaternions and Euler Angles     x A x B x qvqv iq * 2 sin 2 cos                 Rotation around x axis   Ax  1 Ay  1 Az  1 Bz  1 By  1      y A y B y qvqv jq * 2 sin 2 cos                   Ax  1 Ay  1 Az  1 Bz  1 Bx  1  Rotation around y axis     z A z B z qvqv kq * 2 sin 2 cos                   Ax  1 Ay  1 Az  1 Bx  1 By  1 Rotation around z axis
190 ROTATIONS Quaternions (continue – 9) SOLO Description of Successive Rotations Using Quaternions Let describe two consecutive rotations: - First rotation defined by the quaternion                      1 11 1101 ˆ 2 sin, 2 cos, nqq    - Folowed by the second rotation defined by the quaternion                      2 22 2202 ˆ 2 sin, 2 cos, nqq    After the first rotation the quaternion of the vector is transferred to 1 * 1 qvq After the second rotation we obtain      21 * 2121 * 1 * 221 * 1 * 2 qqvqqqqvqqqqvqq  Therefore the quaternion representing those two rotation is:                                                                21 21 1 2 2 1 21 2121 21120210212010220110210 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cos,ˆˆ 2 sin 2 sin 2 cos 2 cos ,,,, nnnnnn qqqqqqqqqq       210 , qqqq    21 2121 0 ˆˆ 2 sin 2 sin 2 cos 2 cos 2 cos nnq                                 21 21 1 2 2 1 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cosˆ 2 sin nnnnn                                  
191 ROTATIONS Quaternions (continue – 10) SOLO Description of Successive Rotations Using Quaternions (continue – 1)    210 , qqqq    21 2121 0 ˆˆ 2 sin 2 sin 2 cos 2 cos 2 cos nnq                                 21 21 1 2 2 1 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cosˆ 2 sin nnnnn                                   Two consecutive rotations, followed by , are given by:1q 2q From those equations we can see that: 0ˆˆˆˆˆˆˆˆˆˆ 21212112211221   nnnnnnnnnnifonlyandifqqqq The rotations are commutative if and only if are collinear.21 ˆ&ˆ nn In matrix form those two rotations are given by: First Rotation:         111111331133 sinˆcos1ˆˆcosˆ,   nnnInR T xx Second Rotation:         222222332233 sinˆcos1ˆˆcosˆ,   nnnInR T xx Total Rotation:              sinˆcos1ˆˆcosˆ,ˆ,ˆ, 331133223333  nnnInRnRnR T xxxx
192 ROTATIONS Quaternions (continue – 11) SOLO Description of Successive Rotations Using Quaternions (continue – 2) Let find the quaternion that describes the Euler Rotations through the angles respectively. Let write the rotations according to their order 123   ,,                                                        2 sin 2 cos 2 sin 2 cos 2 sin 2 cos  ijkqqqq xyz B A                                                                           2 sin 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 sin 2 cos  kjik                                             2 sin 2 sin 2 sin 2 cos 2 cos 2 cos                                             2 cos 2 sin 2 sin 2 sin 2 cos 2 cos  i                                             2 cos 2 sin 2 cos 2 sin 2 cos 2 sin  j                                             2 sin 2 sin 2 cos 2 cos 2 cos 2 sin  k 
193 ROTATIONS Quaternions (continue – 12) SOLO Differential Equation of the Quaternions Let define      ,0qtq B A  - the quaternion that defines the position of B frame relative to frame A at time t.    tqqttq B A    ,00 - the quaternion that defines the position of B frame relative to frame A at time t+Δt.                        t B A ntq ˆ 2 sin, 2 cos  - the quaternion that defines the position of B frame at time t+Δt relative to frame B at time t. We have the relation:      tqtqttq B A B A B A  or                                            ,,0,1,,,,,ˆ 2 sin, 2 cos 00000000 * qqqqqqqqttqtqntq B A B At B A therefore                         tnqq ˆ 2 sin,1 2 cos,, 00                             tnqq ˆ 2 sin,1 2 cos,, 00    or
194 ROTATIONS Quaternions (continue – 13) SOLO Differential Equation of the Quaternions (continue – 1)                          tnqq ˆ 2 sin,1 2 cos,, 00    Let divide both sides by and take the limit .0 tt                                                                                        tBttB t ntqnqn tt tq td d tt q ˆ 2 1 ,0ˆ 2 1 ,0,ˆ 2 2 sin 2 1 , 2 1 2 cos 2 1 ,lim 0 0 0           But is the instant angular velocity vector of frame B relative to frame A.tn ˆ     t B AB nt   ˆ            ttn B AB B ABt     ,0ˆ,0 So we can write        ttqtq td d B AB B A B A   2 1 This is the Differential equation of the quaternion that defines the position of B relative to A, at the time t as a function of the angular velocity vector of frame B relative to frame A, .  tq B A    t B AB 
195 ROTATIONS Quaternions (continue – 14) SOLO Differential Equation of the Quaternions (continue – 2) Developing this equation, we get        ttqtq td d B AB B A B A   2 1               B AB B AB B AB B AB qtq dt d dt dq            00 0 , 2 1 ,0, 2 1 , from which  B AB dt dq    2 10      B AB B ABq dt d       0 2 1 or in matrix form        t Iq q dt d B AB x T                                   330 0 2 1
196 ROTATIONS Quaternions (continue – 15) SOLO Differential Equation of the Quaternions (continue – 3)                                                   zBAB yBAB xBAB qqq qqq qqq qqq q q q q dt d    012 103 230 321 3 2 1 0         t Iq q dt d B AB x T                                   330 0 2 1 B AAB xBAByBABzBAB xBABzBAByBAB yBABzBABxBAB zBAByBABxBAB q q q q q q q q q dt d                                                              2 1 0 0 0 0 2 1 3 2 1 0 3 2 1 0            After rearranging or  zBAByBABxBAB qqq dt dq    321 0 2 1  zBAByBABxBAB qqq dt dq    230 1 2 1  zBAByBABxBAB qqq dt dq    103 2 2 1  zBAByBABxBAB qqq dt dq    012 3 2 1
197 ROTATIONS Quaternions (continue – 16) SOLO Pre-multiply the equation Computation of as a Function of the Quaternion and its Derivatives    t B AB         t Iq q dt d B AB x T                                   330 0 2 1 by          330 xIq                                                t Iq Iq q Iq B AB x T xx                   330 330 0 330 2 1                        ttIIq tIq B BA B BAx TT x T B BAx T         2 1 2 1 2 1 3333 2 0 33 2 0 Therefore                                 0 3302 q Iqt x B AB
198 ROTATIONS Quaternions (continue – 17) SOLO Computation of as a Function of the Quaternion and its Derivatives (continue – 1) But and are related. Differentiating the equation    t B AB  we obtain  0q    1 2 0   T q                                                 3300 0 330 22 xx B AB Iqq q Iqt                          0 033 2 0 330 0 2 1 2 q qIq Iq q T x x T From the equation     TT q qqq 0 000 1 0 we obtain             T x B AB qIq q 033 2 0 0 2
199 ROTATIONS Quaternions (continue – 18) SOLO Computation of as a Function of , and their Derivatives   t B AB   nˆ Differentiate the quaternion                      nqq ˆ 2 sin, 2 cos,0    to obtain                                 nnqq ˆ 2 sinˆ 2 cos 2 , 2 sin 2 ,0     Substitute this in the equation                                 0 3302 q Iqt x B AB                                                                nn nIn x ˆ 2 sinˆ 2 cos 2 2 sin 2 ˆ 2 sin 2 cosˆ 2 sin2 33                                                        nnnnnnn ˆˆ 2 sin2ˆˆ 2 cos 2 sinˆ 2 cos 2 sin2ˆ 2 cosˆ 2 sin 222               nnnnAB ˆˆcos1ˆsinˆ   Finally we obtain We recovered a result obtained before.
200 ROTATIONS Quaternions (continue – 18) SOLO Differential Equation of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I The relations between the components of a vector in the frames A, B and I arev    AB A IA I B A IB I B vCvCCvCv   Using quaternions the same relations are given by   B A A I IA I B A B I IB I B qqvqqqvqv ***  Therefore B A A I B I qqq  B I A I B A qqq *  Let perform the following calculations B A A I B A A I B I q dt d qqq dt d q dt d  & B IB B I B I qq dt d   2 1  A IA A I A I qq dt d   2 1 and use     B A A I B A A IA A I B IB B I q dt d qqqq    2 1 2 1     B A A IA A I A I B IB B I A I B A qqqqqq dt d     1 ** 2 1 2 1 to obtain     B A A IA B IB B A B A qqq dt d    2 1 2 1
201 ROTATIONS Quaternions (continue – 19) SOLO Differential Equatio of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I (continue – 1) Using the relations ABIAIB     and     B A A IA B A B IA qq    *     B A A IA B IA B A qq   we have                  0 2 1 2 1 2 1 2 1 2 1 B A A IA B IA B A B AB B A B A A IA B IA B AB B A B A qqqqqq dt d    from which     B A A AB B AB B A B A qqq dt d    2 1 2 1 Since BAAB     we get         B A A BA B BA B A B A A AB B AB B A B A qqqqq dt d    2 1 2 1 2 1 2 1 From we get1 *  B A A B B A B A qqqq A B B A qq  * Therefore              B A A B B A A B q dt d qqq dt d 0 *B A B A A B A B qq dt d qq dt d             Table of Contents
202 SOLO Laplace Fields Vector Analysis A vector field is said to be a Laplace Field if rAA     0 rA  In this case we have and       022 00 2         AAAAA   0 rA  Harmonic Functions A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions Pierre-Simon Laplace 1749-1827 022             SS dS n dS n 2  0 0 2 1     V GAUSS SS dvdSdS n   Proof: 1 0   S dS n  Proof:   0 0 2 0 2                     dvdS nnS       n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   2nd Green’s Identity:
203 SOLO Vector Analysis Harmonic Functions (continue 1) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 1)  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   3 A function φ harmonic in a volume V can be expressed in terms of the function and its normal derivative on the surface S bounding V. Proof: Use the solution of the Poisson (Laplace) Equation: 02                             S SFSF F dS rrnnrr T r 11 4      where VoutsidendSndS SonF VinF T        11 2 1 1
204 SOLO Vector Analysis Harmonic Functions (continue 2) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 2) RS dSn  1 V Fr  Sr  F SF rrR   4 If the surface S is a sphere SR of radius R with center at then    RS RF dS R r    2 4 1 Fr  Proof: 0 11         SS SF dS nR dS nrr  1 2 11 Rrrn RS SF            Therefore:                            RS R S SFSF F dS R dS rrnnrr T r       2 4 111 4  3 5 If φ is harmonic in a volume V bounded by the surface S and if φ = c = constant at every point on S, then φ = c at every point of V. Proof:     cdc rr dS cdS rrnnrr r S SFS rr SFcSF F SF                                                          4 2 /1 0 4 1 4 111 4 1 2     3
205 SOLO Vector Analysis Harmonic Functions (continue 3) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 3) 6 A non-constant function φ harmonic in a region V can have neither a maximum nor a minimum in V. Proof: For any point inside the region V we can choose an infinitesimal sphere δS centered at this point for which         n n dvdSdS n VSS 100 0      Since can not be either positive or negative inside the region V, the maximum or minimum of the potential φ can occur only at boundary of the region. n  S dSn  1 V Fr  Sr  SF rrr   SF
206 SOLO Vector Analysis Harmonic Functions (continue 4) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 4) 7 If φ is harmonic in a region V bounded by a surface S and ∂ φ/∂ n = 0 at every point of S, then φ = constant at every point of V. S dSn  1 V Fr  Sr  FSF rrr   0   S n  Proof:        SVV dSdvdv  2 Use: 0& 2               SV dSdv  0 2    0 0 2      SV dS n dv   Therefore: VinconstVin .0   1st Green’s Identity:
207 SOLO Vector Analysis Harmonic Functions (continue 5) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 4) 8 If φ1 and φ2 are two solutions of Laplace’s equation in a volume V whose normal derivatives take the same value ∂ φ1/∂ n = ∂ φ2/∂ n on the surface S bounding V, then φ1 and φ2 can differ only by a constant. S dSn  1 V Fr  Sr  FSF rrr   0   S n  Proof: Define: 0: 2 2 1 22 21   021          SSS nnn  From it follows that φ is a constant.7 Table of Contents
208 SOLO Fundamental Theorem of Vector Analysis for a Bounded Region V (Helmholtz’s Theorem) Vector Analysis Hermann Ludwig Ferdinand von Helmholtz 1821 - 1894 Let be a continuous vector field with continuous divergence and curl, in a region V bounded by a surface S. Then has a unique representation as sum of a potential field and a solenoidal field , i.e.  rAA   1 A    2 A A           FFFFF rArArA 21  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr                  S FS S V FS SS V FS S F dS rr rA dv rr rA dv rr rA          4 1 4 1 4 1 :                        S FS S V FS SS V FS SF dS rr rA dv rr rA dv rr rA         4 1 4 11 4 1 :                                         S FS S V FS SS F S FS S V FS SS FF dS rr rA dv rr rA dS rr rA dv rr rA rA            4 1 4 1 4 1 4 1 Therefore
209 SOLO Proof of the Fundamental Theorem of Vector Analysis for a Bounded Region V Vector Analysis Let use the fact that (see GREEN’s FUNCTION):  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr     zzyyxxr SSSS 111    zzyyxxr FFFF 111  where We can write:           z z y y x x SSS S 111 Sr  We define the operator that accts only on . The operator acts only on .           z z y y x x FFF F 111 Fr   SF FS F rr rr              4 12                     1 0 00 dxx x x x                                 V FS SF V FS FS V SFSF dv rr rAdv rr rAdvrrrArA     1 4 11 4 1 22   Using the identity we obtain:  2 FFFFFF                            V FS SFF V FS S FFF dv rr rAdv rr rA rA      1 4 1 4 1 
210 SOLO Vector Analysis Let develop first the divergence expression:  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr                                                                    S FS S V SS FS V FS S S V SS FS V FS SS V FS FS V FS SF dS rr rA dvrA rr dv rr rA dvrA rr dv rr rAdv rr rAdv rr rA                  4 11 4 1 4 11 4 1 1 4 11 4 11 4 1 Define:                       S FS S V SS FSV FS SF dS rr rA dvrA rr dv rr rA        4 11 4 11 4 1 :                           V FS SFF V FS S FFF dv rr rAdv rr rA rA      1 4 1 4 1  Proof of the Fundamental Theorem of Vector Analysis for a Bounded Region V (continue – 1)
211 SOLO Vector Analysis Let develop now the rotor expression:  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Define:                S FS S V FS SS V FS S F dS rr rA dv rr rA dv rr rA          4 1 4 1 4 1 :                                                V FS S S V FS SS V FS SS V FS FS V FS S F dv rr rA dv rr rA dv rr rAdv rr rAdv rr rA             4 1 4 1 1 4 11 4 1 4 1 but                                                  S FS S S FS S S FS S V FS S S tconsC V FS S S dS rr rA CdS rr rA C dS rr rA Cdv rr rA Cdv rr rA C                   4 1 4 1 4 1 4 1 4 1                   S FS S GAUSS V FS S S dS rr rA dv rr rA      4 1 4 1 5                           V FS SFF V FS S FFF dv rr rAdv rr rA rA      1 4 1 4 1  Gauss 5 Proof of the Fundamental Theorem of Vector Analysis for a Bounded Region V (continue – 2)
212 SOLO Vector Analysis We found:  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Therefore:                S FS S V FS SS V FS S F dS rr rA dv rr rA dv rr rA          4 1 4 1 4 1 :                           V FS SFF V FS S FFF dv rr rAdv rr rA rA      1 4 1 4 1                        S FS S V SS FSV FS SF dS rr rA dvrA rr dv rr rA        4 11 4 11 4 1 :    FFF rA   q.e.d. Proof of the Fundamental Theorem of Vector Analysis for a Bounded Region V (continue – 3) Table of Contents
213 SOLO Fundamental Theorem of Vector Analysis for an Unbounded Region (Helmholtz’s Theorem) Vector Analysis For a Bounded region V we found: Hermann Ludwig Ferdinand von Helmholtz 1821 - 1894 Let be a continuous vector field with continuous divergence and curl, such that falls off at infinity like 1/r 1+ε while and fall off at infinity like 1/r 2+ε where ε > 0. Then has a unique representation (to within constant vectors) at sum of a potential field and a solenoidal field , i.e.  rAA   A  A   A   A   rAA  11   rAA  22        0&0 4 1 4 1                                 UU Udv rr rA dv rr rA rA FF V FS SS F V FS SS FF                                          S FS S V FS SS F S FS S V FS SS FF dS rr rA dv rr rA dS rr rA dv rr rA rA            4 1 4 1 4 1 4 1 3/4&4 32 RVRSRrr FS    For an Unbounded region V: The surface integrals are finite only if falls off at infinity like 1/r 1+ε where ε > 0. rA  The volume integrals are finite only if and fall off at infinity like 1/r 2+ε.A   A  
214 SOLO Fundamental Theorem of Vector Analysis for an Unbounded Region (Helmholtz’s Theorem) Vector Analysis     0,0 11  rArA  Therefore Hermann Ludwig Ferdinand von Helmholtz 1821 - 1894 Let be a continuous vector field with continuous divergence and curl, such that falls off at infinity like 1/r 1+ε while and fall off at infinity like 1/r 2+ε where ε > 0. Then has a unique representation (to within constant vectors) at sum of a potential field and a solenoidal field , i.e.  rAA   A  A   A   A   rAA  11   rAA  22        0&0 4 1 4 1                                 UU Udv rr rA dv rr rA rA FF V FS SS F V FS SS FF                                          S FS S V FS SS F S FS S V FS SS FF dS rr rA dv rr rA dS rr rA dv rr rA rA            4 1 4 1 4 1 4 1 Table of Contents
215 REYNOLDS’ TRANSPORT THEOREM This is a part of the Presentations “FLUID DYNAMICS” v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or,  - any system of coordinatesOxyz - any continuous and differentiable functions in    trtr OO ,,, ,,    tandrO,   trO ,,   - flow density at point and time t Or,  SOLO - mass flow through the element .mdsdVS    , sd  - any control volume, changing shape, bounded by a closed surface S(t)v (t) - flow velocity, relative to O, at point and time t trV OOflow ,,,  Or,  - position and velocity, relative to O, of an element of surface, part of the control surface S(t). OSOS Vr ,, ,  - area of the opening i, in the control surface S(t).iopenS - gradient operator in O frame.O, - flow relative to the opening i, in the control surface S(t).OSiOflowSi VVV ,,,   - differential of any vector , in O frame. O td d     FLUID DYNAMICS FLUID DYNAMICS MATHEMATICS SOLO HERMELIN Updated: 5.03.07
216 Start with LEIBNIZ THEOREM from CALCULUS:        ChangeBoundariesthetodueChange tb ta tb ta td tad ttaf td tbd ttbfdx t txf dxtxf td d LEIBNITZ         )),(()),(( ),( ),(:: )( )( )( )(   and generalized it for a 3 dimensional vector space on a volume v(t) bounded by the surface S(t). Using LEIBNIZ THEOREM followed by GAUSS THEOREM (GAUSS 4):                   tv OSOOOSGAUSS OpointtoRelative dsofMovement thetodueChange tS OS tv O LEIBNITZ Otv vdVV t GAUSS sdVvd t vd td d ,,,,)4( )( ,               This is REYNOLDS’ TRANSPORT THEOREM OSBORNE REYNOLDS 1842-1912 SOLO GOTTFRIED WILHELM von LEIBNIZ 1646-1716 REYNOLDS’ TRANSPORT THEOREM v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or,  FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE)
217 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION               )( ,,,,)4( , )()()( tv OSOOOS O GAUSS OS tStv O LEIBNITZ O tv vdVV t GAUSS sdVvd t vd td d                        )( , , )4( , )()()( tv k kOS i k i kOS i GAUSS kkOS tS i tv i LEIBNITZ tv i vd x V x V t GAUSS sdVvd t vd td d            SOLO v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or, 
218 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION O OOS td Rd uV    ,, CASE 1 (CONTROL VOLUME vF ATTACHED TO THE FLUID) kkOS uV ,               )( ,,,)4( , )()()( tv OOO O GAUSS O tStv OO tv F FFF vduu t GAUSS sduvd t vd td d                         )( )4( )()()( tv k k I k I k I GAUSS kK tS I tv I tv I F FFF vd x u x u t GAUSS sduvd t vd td d            SOLO vF (t) SF(t) O x y z sd  OSV ,   0,,,  OSOflowS VVV  OSR , OR,  md OSV ,  OflowV , 
219 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1&,  kkOS uV1&,  uV OS  CASE 2 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )1   )( ,, )( , )( )( tv OO tS O tv F FFF vdusduvd td d td tvd    )()()( )( tv k k k tS k tv F FFF dv x u dsudv td d td tvd                   td tvd tv u F F tv OO F )( )( 1 lim0)( ,,                  td tvd tvx u F F tv k k F )( )( 1 lim0)(  EULER 1755 SOLO vF (t) SF(t) O x y z sd  OSV ,   0,,,  OSOflowS VVV  OSR , OR,  md OSV ,  OflowV , 
220 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CASE 3 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )   &, kkOS uV  &, uV OS    or, since this is true for any attached volume vF(t)             )( ,, )( , )( )( )( 0 tv OO tS O tv tv F FF F vdu t sduvd t vd td d td tmd                      )( )()( )( )( 0 tv k k tS kk tv tv F FF F vdu xt sduvd t dv td d td tmd          Because the Control Volume vF is attached to the fluid and they are not sources or sinks, the mass is constant.   OOOOOO uu t u t ,,,,,, 0           k k k k k x u x u t u xt               0 SOLO vF (t) SF(t) O x y z sd  OSV ,   0,,,  OSOflowS VVV  OSR , OR,  md OSV ,  OflowV , 
221 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,  OSV Define   .... VC OO VC vd t vd td d        .... VC i VC i vd t vd td d                 r t r t r t, , ,        i k k i kx t x t x t, , ,              )( , )()( tS OS tv OO tv sdV vd tt vd td d            k tS kOSi tv i i tv i sdV vd tt vd td d FF           )( , )()(        We have but    OOOO u t u t ,,,, 0            k k iik k u xt u xt             0 CASE 5              r t r t r t, , , SOLO
222 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION We have                                                   )( ,, )( 4 . )( , )( ,,,,,, )( , )( ,, )( tS OOS tv O MDG DerMat GAUSS tS OS tv OOOOOO O tS OS tv OO OO tv sduVvd tD D sdV vduuu t sdV vdu t vd td d                                                                        )( , )( 4 . )( , )( )( , )()( tS kkkOSi tv i MDG DerMat GAUSS tS kkOSi tv k k i k i k k i k i tS kkOSi tv k k i i tv i sduVvd tD D sdV vd x u x u x u t sdV vd x u t vd td d                      CASE 5              r t r t r t, , , SOLO v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or, 
223 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION REYNOLDS 1             )( ,, )( )( tS OOS tv O O tv sduVvd tD D vd td d                   )( , )( )( tS kkkOSi tv i tv i sduVvd tD D dv td d    REYNOLDS 2               )( )( ,, )( tv O tS OSO O tv vd tD D sdVuvd td d                    )( )( , )( tv i tS kkOSki tv i vd tD D sdVuvd td d    CASE 5              r t r t r t, , , SOLO v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or, 
224 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION REYNOLDS 3 CASE 1 (CONTROL VOLUME ATTACHED TO THE FLUID vF(t) ) kkOS uV ,   )()( tv OO tv FF vd tD D vd td d        )()( tv i tv i FF vd tD D vd td d    SOLO O OOS td Rd uV    ,,              r t r t r t, , , vF (t) SF(t) O x y z sd  OSV ,   0,,,  OSOflowS VVV  OSR , OR,  md OSV ,  OflowV ,  CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,  OSV REYNOLDS 4            .. , .. .. SC O O VC VC O sduvd td d vd tD D                 .... .. SC kki VC i VC i sduvd td d vd tD D    Table of Contents
225 Poisson’s Non-homogeneous Differential EquationSOLO The Poisson’s Non-homogeneous Differential Equation for the Static Electric Scalar Potential Ve is: We want to find the Electric Scalar Potential Ve at the point F (field) due to all the sources (S) in the volume V, including its boundaries . n i iSS 1     SFSeS rrrV    1 , 2  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V Therefore is the vector from S to F.SF rrr     zzyyxxr 111    zzyyxxr SSSS 111    zzyyxxr FFFF 111  Let define the operator that acts only on the source coordinate .           z z y y x x SSS S 111 Sr  Note: See development in the Power Point Presentation Table of Contents
226 Poisson’s Non-homogeneous Differential Equation SOLO To find the solution we used the following: • GREEN’s IDENTITY       S ee V ee dSGVVGdvGVVG 22 • GREEN’s FUNCTION This Green’s Function is a particlar solution of the following Poisson’s Non-homogeneous Differential Equation:       0;; 1 ; 2    FSSFS SF FS rrrr rr rrG       FSFSS rrrrG   4;2 Siméon Denis Poisson 1781-1840 GEORGE GREEN 1793-1841
227 Poisson’s Non-homogeneous Differential EquationSOLO • GREEN’s IDENTITY       S ee V ee dSGVVGdvGVVG 22 Let start from the Gauss’ Identity    SV dSAdvA  Karl Friederich Gauss 1777-1855 where is any vector field (function of position and time) continuous and differentiable in the volume V. Let define . A  eVGA     eee VGVGVGA 2   Then        S e V ee V e dSVGdvVGVGdvVG 2 If we interchange G with Ve we obtain        S e V ee V e dSGVdvGVVGdvGV 2 Subtracting the second equation from the first we obtain First Green’s Identity Second Green’s Identity We have
228 SOLO • GREEN’s FUNCTION Define the Green’s Function is a particlar solution of the following Poisson’s Non-homogeneous Differential Equation: where δ (x) is the Dirac function                     1 0 00 dxx x x x   Let use the Fourier Transformation to write                                                                             3 3 3 exp 2 1 exp 2 1 exp 2 1 exp 2 1 exp 2 1 dkrrkj dkdkdkzzkyykxxkj dkzzjkdkyyjkdkxxjk zzyyxxrr SF zyxSFzSFySFx zSFzySFyxSFx SFSFSFSF       zyx zyx dkdkdkdk zkykxkk    3 111  where    FSFSS rrrrG   4;2 Paul Adrien Maurice Dirac 1902 - 1984 Poisson’s Non-homogeneous Differential Equation
229 SOLO • GREEN’s FUNCTION (continue – 1) Let use the Fourier Transformation to write Hence or         SFFS rrkjkgdkrrG  exp; 3            SFSFS rrkjdkrrkjkgdk  exp 2 4 exp 3 3 32              SFSFS rrkjdkrrkjkgdk  exp 2 4 exp 3 3 23   Poisson’s Non-homogeneous Differential Equation Jean Baptiste Joseph Fourier 1768 - 1830
230 SOLO • GREEN’s FUNCTION (continue – 2) Let compute: Therefore: Because this is true for all k, we obtain            SFzSFySFxSSSFS zzkyykxxkjrrkj  expexp2                       SFzSFySFxzyxS zzkyykxxkjzjkyjkxjk exp111        SFzSFySFxSzyx zzkyykxxkjzjkyjkxjk         exp111             SFSFSFS rrkjkrrkjkjkjrrkjkj   expexpexp 2            SFSF rrkjdkrrkjkkgdk  exp 2 4 exp 3 3 23            22 1 2 1 k kg   Poisson’s Non-homogeneous Differential Equation
231 SOLO • GREEN’s FUNCTION (continue – 3) Let use spherical coordinates relative to vector:r                  rr r r kk kk kk z y x z y x 0 0 cos sinsin cossin     dk sin dk dk     dddkkdk sin23    r x y z         SFFS rrkjkgdkrrG  exp; 3          22 1 2 1 k kg   Poisson’s Non-homogeneous Differential Equation
232 SOLO • GREEN’s FUNCTION (continue – 4)        2 3 2 exp 2 1 ; k rkj dkrrG FS           0 0 2 0 2 22 sin cosexp 2 1      dkddk k jkr       0 0 2 coscosexp2 2 1    dkdjkr                      00 0 2 expexp2cosexp1 dk kj jkrjkr r dk jkr jkr        rr dk k kr r 1 2 2sin2 0      Poisson’s Non-homogeneous Differential Equation   2 sin 0    dk k kr where we used (see next slide)   SF FS rrr rrG     11 ;Therefore                 rr r r kk kk kk z y x z y x 0 0 cos sinsin cossin     dk sin dk dk     dddkkdk sin23    r x y z
233 SOLO • GREEN’s FUNCTION (continue – 5) Poisson’s Non-homogeneous Differential Equation     0 sin dk k kr Let compute: x y R  A B C D E F G H Rx Rx  For this use the integral: 0ABCDEFGHA zi dz z e Since z = 0 is outside the region of integration 0    BCDEF ziR xi GHA zi R xi ABCDEFGHA zi dz z e dx x e dz z e dx x e dz z e                  00 0000 sin 2 sin 2 sin lim2limlimlim dk k rk idx x x idx x x idx x ee dx x e dx x e R R R xixi R R xi RR xi R                       idideidei e e dz z e i ii eii i eiez GHA zi      00 1 0 0 00 limlimlim    01 2 2 0 /2 /2sin 0 sin 00        R RRReRii i eRieRz BCDEF zi e R dedededeRi eR e dz z e i ii             Therefore: 0 sin 2 0    idk k rk idz z e ABCDEFGHA zi   2 sin 0    dk k kr
234 Poisson’s Non-homogeneous Differential EquationSOLO • GREEN’s FUNCTION (continue – 6) a Green’s Function for the Poisson’s Non- homogeneous Differential Equation   SF FS rr rrG     1 ; Hence This solution is not unique since we can add any function that satisfies the Laplace’s Equation   0;2  FSS rr  Therefore we have the following Green’s Function       0;; 1 ; 2    FSSFS SF FS rrrr rr rrG    Pierre-Simon Laplace 1749-1827    FSFSS rrrrG   4;2
235 Poisson’s Non-homogeneous Differential EquationSOLO Let return to the Poisson’s Non-homogeneous Differential Equation (1812) for the Electric Scalar Potential Ve is: We want to find the Electric Scalar Potential Ve at the point F (field) due to all the sources (S) in the volume V, including its boundaries  n i iSS 1  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V    SFSeS rrrV    1 , 2  Siméon Denis Poisson 1781-1840
236 Poisson’s Non-homogeneous Differential EquationSOLO Let define the operator that acts only on the source coordinate .Sr            z z y y x x SSS S 111 is the vector from S to F.SF rrr     zzyyxxr 111    zzyyxxr SSSS 111    zzyyxxr FFFF 111  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   0       r r r rr rr rr rr rrr SF SF SF FS SFSS         30  SSSFSS rrrr    0 1 2  r r r rr r dr dG r dr dG rG SS        003 1311 34333 2                    r r r r r r r r r rr r rGrG SSSSSS        SFSeS rrrV    1 , 2  Since is no defined at r = 0 we define the volume V’ as the volume V minus a small sphere of radius and surface around the point F, when F is inside V, or a semi-sphere of radius and surface around the point F, when F is on the boundary of V.  rG 00 r 00 r 2 04 rSF  2 02 rSF 
237 Poisson’s Non-homogeneous Differential EquationSOLO let compute                  '' '0 2 '' 22 , 1 , V SFS V Vin Se V S FS V SeeS dVrrrGdvGVdv r rrGdvGVVG                               FF S SeeS S SeeS dSn r VV r dSGVVG 11                       drn r VV r SeeS r 2 0 11 lim          0 2 0 2 20 0 2 0 0 0 limlimlimlim                     drnVdrn r n VdnVrdnVr Se r e r eS r eS r             VoutsideF SonF VinF rVdrV Fee rr FS 0 2 4 lim     iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Using the Green’s Identity       S ee V ee dSGVVGdvGVVG 22
238 Poisson’s Non-homogeneous Differential EquationSOLO We obtain                                   S e e V SSF dSndS n n S SeeS V SSFFe dS n G V n V G T dvrrrG T dSGVVG T dvrrrG T rV S       4 , 4 4 , 4 1 1   where VoutsidendSndS SonF VinF T        11 2 1 1 Note If F is outside V from the Green’s Second Identity we obtain End Note     VoutsidendSndSdS n G V n V GdSGVVGdvGVVG S e e S SeeS V SeeS               11 22 iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr     SF FS rr rrG     1 ;
239        S D e S DSeFe dS n G V T dSGV T rV  44  where VoutsidendSndS SonF VinF T        11 2 1 1 Poisson’s Non-homogeneous Differential EquationSOLO BOUNDARY CONDITIONS The General Green Function that is a class of bi-position function, and contains an arbitrary harmonic function (solution of the Laplace’s Equation)       0;; 1 ; 2    FSSFS SF FS rrrr rr rrG    Let consider the following two simple cases (Dirichlet and Neumann Problems): 1. Dirichlet Problem Johann Peter Gustav Lejeune Dirichlet 1805-1859 The potential is defined at the boundary S of the volume V.     n i iFe SSongivenrV 1  In this case Let choose such that FS rr  ;   SrrrGG SSFSDirichlet   0; iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
240 Poisson’s Non-homogeneous Differential EquationSOLO BOUNDARY CONDITIONS (continues – 1) 2. Neumann Problem The potential derivative is defined at the boundary S of the volume V. In this case   S S nSSongivennVr n V n i i S eSF e         1&1 1   Let choose such that FS rr  ;     SrnrrG n rrG G S S FSS S FS Neumann        01; ; where VoutsidendSndS SonF VinF T        11 2 1 1        S e N S eSNFe dS n V G T dSVG T rV  44  Franz Neumann 1798-1895 iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
241 Poisson’s Non-homogeneous Differential EquationSOLO Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions Suppose that we have a solution Ve that satisfies the Laplace Homogeneous Differential Equation:   0, 2  FSeS rrV  in the volume V, including its boundaries . n i iSS 1  Suppose also that Dirichlet or Neumann conditions or a combination of those, are specified. In this case the solution is unique (up to an additive constant). Proof Suppose that thee exist two solutions and , and define rVe1  rVe 2      rVrVr ee 21   We have           0 1 2 2 1 1 22     SS r e r e rVrVr    
242 Poisson’s Non-homogeneous Differential EquationSOLO Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions If Dirichlet conditions are satisfied: Proof (continue) Let use Green’s First Identity (with G = Φ) (continue – 1)              n i i rVrV FeFeF SSonrVrVr FeFe 1 21 0 21    If Neumann conditions are satisfied:              n i i r n V r n V F e F e F SSonr n V r n V r n F e F e 1 21 0 21                         SSVV dS n dSGdvdv 2 We have   VinconstVVVindS n dv ee Dirichlet Neumann SV      21 00 End of Proof
243 Poisson’s Non-homogeneous Differential EquationSOLO In the same way the solution of the Poisson’s Non-homogeneous Differential Equation for the Vector Potential is: F rA                                                                          S SF S S SFV SF SS dSndS n n S SF SSSS SFV SF SS F dS rrn rA n rA rr T dv rr rAT dS rr rArA rr T dv rr rAT rA S          11 44 11 44 2 1 1 2   where VoutsidendSndS SonF VinF T        11 2 1 1 iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Table of Contents
244 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION The Helmholtz Non-homogeneous Differential Equation for the Electric Scalar Potential Ve is:      trtrV tv trV eee , 1 , 1 , 2 2 2 2        We want to find the Electric Scalar Potential Ve at the point F (field) due to all the sources (S) in the volume V, including its boundaries . n i iSS 1  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V Therefore is the vector from S to F.SF rrr     zzyyxxr 111    zzyyxxr SSSS 111    zzyyxxr FFFF 111  Let define the operator that acts only on the source coordinate .           z z y y x x SSS S 111 Sr  This is a part of the presentation “Electromagnetics” SOLO HERMELIN ELECTROMAGNETICS
245 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION To find the solution we need to prove the following: • GREEN’s IDENTITY       S ee V ee dSGVVGdVGVVG 22 • GREEN’s FUNCTION   FS FS FS rr v rr tt trtrG                ' ',;,  This Green’s Function is a particular solution of the following Helmholtz Non-homogeneous Differential Equation:        '4',;, 1 ',;, 2 2 2 2 ttrrtrtrG tv trtrG SFFSFSS       (continue – 1) iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V
246 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • Scalar Green’s Identities       S ee V ee dSGVVGdVGVVG 22 (continue – 2) Let start from the Gauss’ Divergence Theorem    SV dSAdVA  Karl Friederich Gauss 1777-1855 where is any vector field (function of position and time) continuous and differentiable in the volume V. Let define . A  eVGA     eee VGVGVGA 2   Then        S e Gauss V ee V e dSVGdVVGVGdVVG 2 If we interchange G with Ve we obtain        S e Gauss V ee V e dSGVdVGVVGdVGV 2 Subtracting the second equation from the first we obtain First Green’s Identity Second Green’s Identity We have GEORGE GREEN 1793-1841
247 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION Define the Green’s Function as a particular solution of the following Helmholtz Non-homogeneous Differential Equation:        '4',;, 1 ',;, 2 2 2 2 ttrrtrtrG tv trtrG SFFSFSS       (continue – 3) where δ (x) is the Dirac function                     1 0 00 dxx x x x   Let use the Fourier Transformation to write                                                                             3 3 3 exp 2 1 exp 2 1 exp 2 1 exp 2 1 exp 2 1 dkrrkj dkdkdkzzkyykxxkj dkzzjkdkyyjkdkxxjk zzyyxxrr SF zyxSFzSFySFx zSFzySFyxSFx SFSFSFSF       zyx zyx dkdkdkdk zkykxkk    3 111  where Paul Dirac 1902-1984 Joseph Fourier 1768-1830
248 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 1) In the same way: (continue – 4)            dttjtt 'exp 2 1 ' Therefore               'expexp 2 1 ' 3 4 ttjrrkjddkttrr SFSF     Let use the Fourier Transformation to write             'expexp,',;, 3 ttjrrkjkgddktrtrG SFFS   Hence                            'expexp 2 4 'expexp, 1 3 4 3 2 2 2 2 ttjrrkjddkttjrrkjkgddk tv SFSFS      or                                  'expexp 2 4 'expexp 1 exp'exp, 3 4 2 2 2 23 ttjrrkjddk ttj t rrkj v rrkjttjkgddk SF SFSFS      
249 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 2) Let compute: (continue – 5)                                         SFSFSFS SFzSFySFxSzyx SFzSFySFxzyxS SFzSFySFxSSSFS rrkjkrrkjkjkjrrkjkj zzkyykxxkjzjkyjkxjk zzkyykxxkjzjkyjkxjk zzkyykxxkjrrkj                           expexpexp exp111 exp111 expexp 2 2      'exp'exp 2 2 2 ttjttj t     Therefore:                            'expexp 2 4 'expexp, 3 4 2 2 23 ttjrrkjddk ttjrrkj v kkgddk SF SF         Because this is true for all k and ω, we obtain                 2 2 2 3 1 4 1 , v k kg   
250 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 3) (continue – 6)                                  'expexp 4 1 'expexp,',;, 2 2 2 3 3 3 ttjrrkj v k d dkttjrrkjkgddktrtrG SFSFFS       We can see that the integral in k has to singular points for v k   Let consider only the progressive wave, i.e. G = 0 for t > t’. To find the integral let change ω by ω + jδ where δ is a small negative number                           rkj v j k d dktrtrG FS      exp 4 1 ',;, 2 2 2 3 3 where and .SF rrr   'tt 
251 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 4) (continue – 7) In the plane ω we close the integration path by the semi-circle with and the singular points on the upper side, for τ > 0 (for t > t’)                           rkj v j k d dktrtrG FS      exp 4 1 ',;, 2 2 2 3 3 r      '00exp ttdjf UPC        '00exp ttdjf DOWNC                                 0exp 0exp exp      DOWN UP C C djf djf djf jvk  jvk  Re Im
252 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 5) (continue – 8)                                                               CC jvkjvk rkjv drkj v j k d rkj v j k d I           exp expexp 2 2 2 2 2 2 2 Let use the Cauchy Integral for a complex function f (z) continuous on a closed path C, in the complex z plane:      0 0 2lim2 0 zfjzfjdz zz zf zz C     We have:                       k kvrkj v vk jkv vk jkv rkjvj jvk rkjv j jvk rkjv jI vkvk            sinexp 2 2 exp 2 exp exp2 exp 2lim exp 2lim 2 2 0, 2 0,                      Therefore, we can write:                             k vkrkj dk v v k rkj ddktrtrG FS      sinexp 2 exp 4 1 ',;, 3 2 2 2 2 3 3  
253 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 6) (continue – 9) Let use spherical coordinates relative to vector:                             k vkrkj dk v v k rkj ddktrtrG FS      sinexp 2 exp 4 1 ',;, 3 2 2 2 2 3 3   r                  rr r r kk kk kk z y x z y x 0 0 cos sinsin cossin     dk sin dk dk     dddkkdk sin23    r x y z
254 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 7) (continue – 10)  kvd v r jkv v r jkv v r jkv v r jkv r                                                      4 expexpexpexp 1   r v rr tt v rr tt SFSF                  ''  r v r v r dx v r jx v r jx r                                                    expexp 2 11    kvd v r jkv v r jkv r v kvd v r jkv v r jkv r                                                          4 expexp 4 expexp 1                               dk j jvkjvk j jkrjkr r v 2 expexp 2 expexp               dkvkvkr r v dkvkvkr r v     sinsinsinsin 2 0                         00 0 sin 2 expexp2cosexp sin dkvk j jkrjkr r v dk jkr jkr vkk v                0 0 2 cossincosexp2 2    dkdvkjkrk v          0 0 2 0 2 2 sin sincosexp 2      dkddk k vkjkrv          k vkrkj dk v trtrG FS   sinexp 2 ',;, 3 2                   rr r r kk kk kk z y x z y x 0 0 cos sinsin cossin     dk sin dk dk     dddkkdk sin23    r x y z
255 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 8) (continue – 11) We can see that represents a progressive wave and represents a regressive wave:                    v rr tt v rr tt SFSF  ''                     v rr tt v rr tt SFSF  ''  Hence   SF SFSF FS rr v rr tt v rr tt trtrG                         '' ',;,  We shall consider only the progressive wave and use:   SF SF FS rr v rr tt trtrG                ' ',;,  Retarded Green Function The other solution is:   SF SF FS rr v rr tt trtrG                ' ',;,  Advanced Green Function
256 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 12) Gustav Robert Kirchhoff 1824-1887 Let return to the Helmholtz Non-homogeneous Differential Equation for the Electric Scalar Potential Ve is:      trtrV tv trV eee , 1 , 1 , 2 2 2 2        We want to find the Electric Scalar Potential Ve at the point F (field) due to all the sources (S) in the volume V, including its boundaries  n i iSS 1  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V Hermann von Helmholtz 1821-1894
257 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 13) Since is no defined at r = 0 we define the volume V’ as the volume V minus a small sphere of radius and surface around the point F, when F is inside V, or a semi-sphere of radius and surface around the point F, when F is on the boundary of V.  rG 00 r 00 r 2 04 rSF  2 02 rSF  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Let define the operator that acts only on the source coordinate .Sr            z z y y x x SSS S 111 Using the Green’s Identity                      FSS SFSSeSeSSF V SFSSeSeSSF dStrtrGtrVtrVtrtrGdVtrtrGtrVtrVtrtrG ',;,',',',;,',;,',',',;, ' 22  substitute here      ', 1 ', ' 1 ', 2 2 2 2 trtrV tv trV SeSeSeS               '4',;, ' 1 ',;, 2 2 2 2 ttrrtrtrG tv trtrG FSFSFSS       we obtain                                                   FSS SFSSeeSSSF V SFSe S SF V SFSeSeSF dStrtrGtrVtrVtrtrG dVttrrtrV tr trtrG dVtrtrG t trVtrV t trtrG v ',;,',',',;, '4', ', ',;, ',;, ' ',', ' ',;, 1 ' ' 2 2 2 2 2        
258 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 13) Let integrate the equation between to and choose t such that:1' tt  2' tt  21 /' tvrttt                              2 2 1 1 2 1 ''4', ', ',;, '',;, ' ',', ' ',;, 1 ' ' 2 2 2 2 2 I t t V SFSe S SF I t t V SFSeSeSF dtdVttrrtrV tr trtrG dtdVtrtrG t trVtrV t trtrG v                                                  4 2 1 3 2 1 '',;,',',',;, '',;,',',',;, I t t S SFSSeeSSSF I t t S SFSSeeSSSF dtdStrtrGtrVtrVtrtrG dtdStrtrGtrVtrVtrtrG F       
259 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 14) Integral I1     0 ' ',;,',;, 21            SF SF SFSF rr v rr tt ttrtrGttrtrG     Since 21 /' tvrttt  then                         0',;, ' ',', ' ',;, 1 '',;, ' ',', ' ',;, ' 1 '',;, ' ',', ' ',;, 1 ' 2 ' 2 ' 2 2 2 2 21 2 1 2 1 2 1                                           V t t SFSeSeSF V t t SFSeSeSF t t V SFSeSeSF dVtrtrG t trVtrV t trtrG v dVdttrtrG t trVtrV t trtrG tv dVdttrtrG t trVtrV t trtrG v I        0',;, ' ',;, ' 21       ttrtrG t ttrtrG t SFSF 
260 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 15) Integral I2                  2 1 ''4', ', ',;, ' 2 t t V SFSe S SF dtdVttrrtrV tr trtrGI                               2 1 2 1 '' ''',4' ',/' t t t t V SFSe V S FS dVdtttrrtrVdtdV tr rr vrtt                        ' /' 0 '' /' ',1 '',4 ',1 V vrttFS S V SFSe V vrttFS S dV rr tr dVttrrtrVdV rr tr              The integral is zero since in V’.     ' /', V FSSe dVrrvrttrV   FS rr   iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
261 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 16) Integral I3            2 1 '',;,',',',;,3 t t S SFSSeSeSSF dSdttrtrGtrVtrVtrtrGI                                              S t t SF SF SSeSeS SF SF dSdt rr v rr tt trVtrV rr v rr tt2 1 ' ' ',', '                                                  S t t SF S SF SeSeS SF SF dSdt rrv rr tttrVtrV rr v rr tt2 1 ' 1 '',', '                                      S t t SF SF S Se dSdt rr v rr tt trV 2 1 ' ' ',    
262 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 17) Integral I3 (continue – 1) But     v rr vrtt t rrvrtt rv rr tt SFS SFS rrr SF S SF                       /' ' /''  and                                                       S t t SF SF SFS Se S t t SF SF S Se dSdt v rr tt trrv rr trVdSdt rr v rr tt trV 2 1 2 1 '' ' ',' ' ',                                                   S t t SF SF SFS Se S t t SF SF SFS Se partsbyegrating dSdt v rr tt rrv rr trV t dS v rr tt rrv rr trV 2 1 2 1 ''', ' '', 0 int            
263 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 18) Integral I3 (continue – 2) Therefore                                             S t t SF Se SF SFS SF SSe SF SeS dSdt v rr tttrV trrv rr rr trV rr trV I 2 1 ''', ' 1 ', ', 3                                             S vrtt Se SF SFS SF SSe SF SeS dStrV trrv rr rr trV rr trV /' ', ' 1 ', ',                                                S vrtt Se SF SF SF SSe SF Se dStrV trrv rr n rr trV rr n trV /' ', ' 1 ', ',        The last equality follows from dS n U dSnn n U dSU SS             11
264 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 19) Integral I4 In the same way as for integral I4 we have             2 1 '',;,',',',;,4 t t S SFSSeSeSSF F dSdttrtrGtrVtrVtrtrGI                                     FS vrtt Se SF SFS SF SSe SF SeS dStrV trrv rr rr trV rr trV /' ', ' 1 ', ',                                                FS vrtt Se SF SF SF SSe SF Se dStrV trrv rr n rr trV rr n trV /' ', ' 1 ', ',       
265 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 20) Integral I4 (continue – 1) We use since points inside V normal to and points outside V.   n r r 1 0 0  0 0 r r  nS  n1 On the sphere or the semi-spherearound the field point F with radius and surface or if the point F is inside V or on the boundary, respectively, we have  FS rrr   00 2 04 rSF  2 02 rSF          n r r rr rr rr rr rr SF SF SF FS SSFS F 1 0 0                   n rr r rrr rr rrrr rr rrrr SF SF SFSF FS SFSSF S F 1 11111 2 00 0 2 0 22         ndr r r drdS FS 12 0 0 02 0  Since we can assume mean values for all field quantities in the integral00 r iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
266 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 21) Integral I4 (continue – 2)                                    FS vrtt Se SF SFS SF SSe SF SeS dStrV trrv rr rr trV rr trV I /' 4 ', ' 1 ', ',                              drnn rv trV t n r trV r trV vrtt SeSe SeS r 2 0 /'0 2 00 0 11 1 ', ' 1 1 ', ', lim 0                   0 0 /' 0/'0 0 0/'0 ', ' lim',lim1',lim 000                   d v r trV t dtrVdrntrV vrtt Se rvrttSe rvrttSeS r    trV SonF VinF trV SonFd VinFd FeFe , 2 4 ,2 0 4 0                                  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
267 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 22) SUMMARIZE The Kirchhoff’s solution to the Helmholtz Non-homogeneous Differential Equation:      trtrV tv trV eee , 1 , 1 , 2 2 2 2        is                                             S v rr tt Se SF SFS SF SSe SF SeS V v rr ttSF S Fe dStrV trrv rr rr trV rr trVT dV rr trT trV SFSF            '' ', ' 1 ', ', 4 ', 4 ,                                                         S v rr tt Se SF SF SF Se SF Se V v rr ttSF S dSndS n n dStrV trrv rr n rrn trV rr n trV T dV rr trT SF SF S            ' ' ˆ ˆ ', ' 1 ', ', 4 ', 4    Voutsiden SonF VinF T       1 2 1 1 iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Table of Contents
268 SOLO References M.R. Spiegel, “Vector Analysis and an Introduction to Tensor Analysis”, Schaum’s Outline Series, McGraw-Hill, 1959 Vector Analysis H.Lass, “Vector and Tensor Analysis”, McGraw Hill, 1950 J,N, Reddy & M.L. Rasmussen, “Advanced Engineering Analysis”, John Willey, 1982, Ch.1:”Elements of Vector and Tensor Analysis” Table of Contents
January 6, 2015 269 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

More Related Content

PDF
Chapter 16 1
EasyStudy3
 
PPT
Vector analysis
NUSRAT JAHAN
 
PPT
Vector calculus
Kumar
 
PPTX
Vector calculus
Santhanam Krishnan
 
PPTX
Electromagnetic fields: Review of vector algebra
Dr.SHANTHI K.G
 
PDF
4-scalarsvectors-20250611124200_12345.pdf
arif69389
 
PPTX
Coordinate system
MOHAMMAD AKRAM
 
PPTX
VECTOR CALCULUS
MANJULAKAMALANATHAN
 
Chapter 16 1
EasyStudy3
 
Vector analysis
NUSRAT JAHAN
 
Vector calculus
Kumar
 
Vector calculus
Santhanam Krishnan
 
Electromagnetic fields: Review of vector algebra
Dr.SHANTHI K.G
 
4-scalarsvectors-20250611124200_12345.pdf
arif69389
 
Coordinate system
MOHAMMAD AKRAM
 
VECTOR CALCULUS
MANJULAKAMALANATHAN
 

What's hot (20)

PPS
Triple product of vectors
guest581a478
 
PPTX
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...
Smit Shah
 
PPTX
Line integeral
Tayyaba Abbas
 
PDF
Hamilton application
Samad Akbar
 
PPTX
Diagonalization of Matrices
AmenahGondal1
 
PDF
Vector differentiation, the ∇ operator,
Tarun Gehlot
 
PPTX
Runge kutta
Shubham Tomar
 
PPTX
Differential equations
Muhammad Ali Bhalli Zada
 
PPTX
Gradient , Directional Derivative , Divergence , Curl
VishalVishwakarma59
 
PDF
Succesive differntiation
JaydevVadachhak
 
PPTX
Tensor analysis
University of Education
 
PPTX
CURVATURE AND RADIUS OF CURVATURE
AbhishekRan
 
PPTX
Dot & cross product of vectors
Ashraful Tauhid
 
PPTX
Scrodinger wave equation
Sujit K Shah, Ph D
 
PPTX
presentation on Euler and Modified Euler method ,and Fitting of curve
Mukuldev Khunte
 
DOCX
Application of vector integration
Varuna Kapuge
 
PPTX
GAUSS ELIMINATION METHOD
reach2arkaELECTRICAL
 
PPTX
Analytic function
Santhanam Krishnan
 
PPTX
Verification of Solenoidal & Irrotational
MdAlAmin187
 
PPT
systems of linear equations & matrices
Student
 
Triple product of vectors
guest581a478
 
ORTHOGONAL, ORTHONORMAL VECTOR, GRAM SCHMIDT PROCESS, ORTHOGONALLY DIAGONALI...
Smit Shah
 
Line integeral
Tayyaba Abbas
 
Hamilton application
Samad Akbar
 
Diagonalization of Matrices
AmenahGondal1
 
Vector differentiation, the ∇ operator,
Tarun Gehlot
 
Runge kutta
Shubham Tomar
 
Differential equations
Muhammad Ali Bhalli Zada
 
Gradient , Directional Derivative , Divergence , Curl
VishalVishwakarma59
 
Succesive differntiation
JaydevVadachhak
 
Tensor analysis
University of Education
 
CURVATURE AND RADIUS OF CURVATURE
AbhishekRan
 
Dot & cross product of vectors
Ashraful Tauhid
 
Scrodinger wave equation
Sujit K Shah, Ph D
 
presentation on Euler and Modified Euler method ,and Fitting of curve
Mukuldev Khunte
 
Application of vector integration
Varuna Kapuge
 
GAUSS ELIMINATION METHOD
reach2arkaELECTRICAL
 
Analytic function
Santhanam Krishnan
 
Verification of Solenoidal & Irrotational
MdAlAmin187
 
systems of linear equations & matrices
Student
 
Ad

Viewers also liked (20)

PDF
Shadechapter12.ppt [read only]
betomedic
 
PPT
Matrices i
Solo Hermelin
 
PDF
Complex analysis book by iit
JITENDRA SUWASIYA
 
DOCX
15 bcm0081 assign 2
Madhupraveen Peesapati
 
PPSX
Web analytics and matrics basics
JOSH talks
 
PDF
AP Calculus Slides March 22, 2007
Darren Kuropatwa
 
PPT
MATRICS – Sequestration and Trust Deeds
shelterscotland
 
PDF
Shadechapter13.ppt [read only]
betomedic
 
PPT
Introduction to Mathematical Probability
Solo Hermelin
 
PDF
Golden words of swami vivekananda
chiefs world
 
PPTX
Basic Integral
Daffodil International University
 
PPT
Legendre's eqaution
Sachin Patel
 
PPSX
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...
Alex Bell
 
PPT
Swami Vivekananda Quotes
Moncy Varghese
 
PPSX
Lecture notes for s4 b tech Mathematics
Anoop T Vilakkuvettom
 
PPT
Prime numbers
Solo Hermelin
 
PPTX
Bessel’s equation
kishor pokar
 
PDF
Swami vivekananda’s 150 quotes
Firoj Md Shah
 
PPTX
Inverse matrix
Melody Kaye
 
PPT
Practical Applications of Bessel's function
OneirosErebus
 
Shadechapter12.ppt [read only]
betomedic
 
Matrices i
Solo Hermelin
 
Complex analysis book by iit
JITENDRA SUWASIYA
 
15 bcm0081 assign 2
Madhupraveen Peesapati
 
Web analytics and matrics basics
JOSH talks
 
AP Calculus Slides March 22, 2007
Darren Kuropatwa
 
MATRICS – Sequestration and Trust Deeds
shelterscotland
 
Shadechapter13.ppt [read only]
betomedic
 
Introduction to Mathematical Probability
Solo Hermelin
 
Golden words of swami vivekananda
chiefs world
 
Basic Integral
Daffodil International University
 
Legendre's eqaution
Sachin Patel
 
Complex Analysis - Differentiability and Analyticity (Team 2) - University of...
Alex Bell
 
Swami Vivekananda Quotes
Moncy Varghese
 
Lecture notes for s4 b tech Mathematics
Anoop T Vilakkuvettom
 
Prime numbers
Solo Hermelin
 
Bessel’s equation
kishor pokar
 
Swami vivekananda’s 150 quotes
Firoj Md Shah
 
Inverse matrix
Melody Kaye
 
Practical Applications of Bessel's function
OneirosErebus
 
Ad

Similar to Vector analysis (20)

PPT
Dyadics
Solo Hermelin
 
PPTX
Vectors mod-1-part-1
Asmita Bhagdikar
 
PPTX
Vectorspace in 2,3and n space
Ahmad Saifullah
 
PDF
Vectouurs
junaid novapex
 
PDF
Vectouurs
junaid novapex
 
ODP
Lesson 1: Vectors and Scalars
VectorKing
 
PDF
Chapter 1
Tehulu Belay
 
PDF
Lines and planes in space
Tarun Gehlot
 
PPT
Vectors Victor
Neil MacIntosh
 
PDF
A some basic rules of tensor calculus
Tarun Gehlot
 
PPT
Differential geometry three dimensional space
Solo Hermelin
 
PPT
1638 vector quantities
Dr Fereidoun Dejahang
 
PPTX
vector and tensor.pptx
Goutampodder3
 
PDF
Lecture2 (vectors and tensors).pdf
entesarkareem1
 
PPTX
Electric and Magnetic Fields (EEE2303)-lecture 1-3 - Vector Analysis.pptx
monaibrahim598401
 
DOCX
SPHA021 Notes-Classical Mechanics-2020.docx
lekago
 
DOC
1.1 types of vectors (1)
ÀŚhü Řąną
 
PPTX
1-0 Vectors (1).pptxhesagdfhjwegsdfjhsdbcjhew
zqweo23
 
PPTX
Beginning direct3d gameprogrammingmath03_vectors_20160328_jintaeks
JinTaek Seo
 
PDF
Ec8451 - Electro Magnetic Fields
Benesh Selvanesan
 
Dyadics
Solo Hermelin
 
Vectors mod-1-part-1
Asmita Bhagdikar
 
Vectorspace in 2,3and n space
Ahmad Saifullah
 
Vectouurs
junaid novapex
 
Vectouurs
junaid novapex
 
Lesson 1: Vectors and Scalars
VectorKing
 
Chapter 1
Tehulu Belay
 
Lines and planes in space
Tarun Gehlot
 
Vectors Victor
Neil MacIntosh
 
A some basic rules of tensor calculus
Tarun Gehlot
 
Differential geometry three dimensional space
Solo Hermelin
 
1638 vector quantities
Dr Fereidoun Dejahang
 
vector and tensor.pptx
Goutampodder3
 
Lecture2 (vectors and tensors).pdf
entesarkareem1
 
Electric and Magnetic Fields (EEE2303)-lecture 1-3 - Vector Analysis.pptx
monaibrahim598401
 
SPHA021 Notes-Classical Mechanics-2020.docx
lekago
 
1.1 types of vectors (1)
ÀŚhü Řąną
 
1-0 Vectors (1).pptxhesagdfhjwegsdfjhsdbcjhew
zqweo23
 
Beginning direct3d gameprogrammingmath03_vectors_20160328_jintaeks
JinTaek Seo
 
Ec8451 - Electro Magnetic Fields
Benesh Selvanesan
 

More from Solo Hermelin (20)

PPT
5 introduction to quantum mechanics
Solo Hermelin
 
PPT
Stabilization of linear time invariant systems, Factorization Approach
Solo Hermelin
 
PPT
Slide Mode Control (S.M.C.)
Solo Hermelin
 
PPT
Sliding Mode Observers
Solo Hermelin
 
PPT
Reduced order observers
Solo Hermelin
 
PPT
Inner outer and spectral factorizations
Solo Hermelin
 
PPT
Keplerian trajectories
Solo Hermelin
 
PPT
Anti ballistic missiles ii
Solo Hermelin
 
PPT
Anti ballistic missiles i
Solo Hermelin
 
PPT
Analytic dynamics
Solo Hermelin
 
PPT
12 performance of an aircraft with parabolic polar
Solo Hermelin
 
PPT
11 fighter aircraft avionics - part iv
Solo Hermelin
 
PPT
10 fighter aircraft avionics - part iii
Solo Hermelin
 
PPT
9 fighter aircraft avionics-part ii
Solo Hermelin
 
PPT
8 fighter aircraft avionics-part i
Solo Hermelin
 
PPT
6 computing gunsight, hud and hms
Solo Hermelin
 
PPT
4 navigation systems
Solo Hermelin
 
PPT
3 earth atmosphere
Solo Hermelin
 
PPT
2 aircraft flight instruments
Solo Hermelin
 
PPT
3 modern aircraft cutaway
Solo Hermelin
 
5 introduction to quantum mechanics
Solo Hermelin
 
Stabilization of linear time invariant systems, Factorization Approach
Solo Hermelin
 
Slide Mode Control (S.M.C.)
Solo Hermelin
 
Sliding Mode Observers
Solo Hermelin
 
Reduced order observers
Solo Hermelin
 
Inner outer and spectral factorizations
Solo Hermelin
 
Keplerian trajectories
Solo Hermelin
 
Anti ballistic missiles ii
Solo Hermelin
 
Anti ballistic missiles i
Solo Hermelin
 
Analytic dynamics
Solo Hermelin
 
12 performance of an aircraft with parabolic polar
Solo Hermelin
 
11 fighter aircraft avionics - part iv
Solo Hermelin
 
10 fighter aircraft avionics - part iii
Solo Hermelin
 
9 fighter aircraft avionics-part ii
Solo Hermelin
 
8 fighter aircraft avionics-part i
Solo Hermelin
 
6 computing gunsight, hud and hms
Solo Hermelin
 
4 navigation systems
Solo Hermelin
 
3 earth atmosphere
Solo Hermelin
 
2 aircraft flight instruments
Solo Hermelin
 
3 modern aircraft cutaway
Solo Hermelin
 

Recently uploaded (20)

PPTX
2. Earth - The Living Planet earth and life
markjustinebarolobau
 
PPTX
ECG_Course_Presentation د.محمد صقران ppt
lelouchml1995
 
PDF
Phytochemical Investigation of Miliusa longipes.pdf
IrfanShahirSharafi
 
DOCX
Q1_LE_Mathematics 8_Lesson 5_Week 5.docx
marcusaviso1101
 
PDF
VARICELLA VACCINATION: A POTENTIAL STRATEGY FOR PREVENTING MULTIPLE SCLEROSIS
ijab2
 
PDF
Formation of Supersonic Turbulence in the Primordial Star-forming Cloud
Sérgio Sacani
 
PDF
diccionario toefl examen de ingles para principiante
Karla Montoya
 
PPTX
ANEMIA WITH LEUKOPENIA MDS 07_25.pptx htggtftgt fredrctvg
vedvyas5668
 
PDF
An interstellar mission to test astrophysical black holes
Sérgio Sacani
 
PDF
bbec55_b34400a7914c42429908233dbd381773.pdf
agambirev
 
PPTX
ognitive-behavioral therapy, mindfulness-based approaches, coping skills trai...
OliveJolly1
 
PPT
protein biochemistry.ppt for university classes
bayewodajo27
 
PDF
ELS_Q1_Module-11_Formation-of-Rock-Layers_v2.pdf
savannaesar07
 
PPTX
Taita Taveta Laboratory Technician Workshop Presentation.pptx
CALVINCEOWINO
 
PPTX
2. Earth - The Living Planet Module 2ELS
markjustinebarolobau
 
PDF
. Radiology Case Scenariosssssssssssssss
MiraMusleh
 
PDF
Mastering Bioreactors and Media Sterilization: A Complete Guide to Sterile Fe...
Sabina Sharma
 
PPTX
SCIENCE10 Q1 5 WK8 Evidence Supporting Plate Movement.pptx
ROLANARIBATO3
 
PDF
Unveiling a 36 billion solar mass black hole at the centre of the Cosmic Hors...
Sérgio Sacani
 
PPTX
Protein & Amino Acid Structures Levels of protein structure (primary, seconda...
Muhammad Sajid Afridi
 
2. Earth - The Living Planet earth and life
markjustinebarolobau
 
ECG_Course_Presentation د.محمد صقران ppt
lelouchml1995
 
Phytochemical Investigation of Miliusa longipes.pdf
IrfanShahirSharafi
 
Q1_LE_Mathematics 8_Lesson 5_Week 5.docx
marcusaviso1101
 
VARICELLA VACCINATION: A POTENTIAL STRATEGY FOR PREVENTING MULTIPLE SCLEROSIS
ijab2
 
Formation of Supersonic Turbulence in the Primordial Star-forming Cloud
Sérgio Sacani
 
diccionario toefl examen de ingles para principiante
Karla Montoya
 
ANEMIA WITH LEUKOPENIA MDS 07_25.pptx htggtftgt fredrctvg
vedvyas5668
 
An interstellar mission to test astrophysical black holes
Sérgio Sacani
 
bbec55_b34400a7914c42429908233dbd381773.pdf
agambirev
 
ognitive-behavioral therapy, mindfulness-based approaches, coping skills trai...
OliveJolly1
 
protein biochemistry.ppt for university classes
bayewodajo27
 
ELS_Q1_Module-11_Formation-of-Rock-Layers_v2.pdf
savannaesar07
 
Taita Taveta Laboratory Technician Workshop Presentation.pptx
CALVINCEOWINO
 
2. Earth - The Living Planet Module 2ELS
markjustinebarolobau
 
. Radiology Case Scenariosssssssssssssss
MiraMusleh
 
Mastering Bioreactors and Media Sterilization: A Complete Guide to Sterile Fe...
Sabina Sharma
 
SCIENCE10 Q1 5 WK8 Evidence Supporting Plate Movement.pptx
ROLANARIBATO3
 
Unveiling a 36 billion solar mass black hole at the centre of the Cosmic Hors...
Sérgio Sacani
 
Protein & Amino Acid Structures Levels of protein structure (primary, seconda...
Muhammad Sajid Afridi
 

Vector analysis

  • 2. 2 Vector AnalysisSOLO TABLE OF CONTENT Algebras History Vector Analysis History Vector Algebra Reciprocal Sets of Vectors Vector Decomposition The Summation Convention The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee   Change of Vector Base, Coordinate Transformation Vector Space Differential Geometry Osculating Circle of C at P Theory of Curves Unit Tangent Vector of path C at a point P Curvature of curve C at P Osculating Plane of C at P Binormal Torsion Seret-Frenet Equations Involute Evolute Vector Identities Summary Cartesian Coordinates
  • 3. 3 Vector AnalysisSOLO TABLE OF CONTENT (continue – 1) Differential Geometry Conjugate Directions Surfaces in the Three Dimensional Spaces First Fundamental Form: Arc Length on a Path on the Surface Surface Area Change of Coordinates Second Fundamental Form Principal Curvatures and Directions Asymptotic Lines Scalar and Vector Fields Vector Differentiation Ordinary Derivative of Scalars and Vectors Partial Derivatives of Scalar and Vectors Differentials of Vectors The Vector Differential Operator Del (, Nabla) Scalar Differential Vector Differential Differential Identities
  • 4. 4 Vector AnalysisSOLO TABLE OF CONTENT (continue – 2) Scalar and Vector Fields Curvilinear Coordinates in a Three Dimensional Space Covariant and Contravariant Components of a Vector in Base .321 ,, uuu rrr  Coordinate Transformation in Curvilinear Coordinates Covariant Derivative Covariant Derivative of a Vector .A  Vector Integration Ordinary Integration of Vectors Line Integrals Surface Integrals Volume Integrals Simply and Multiply Connected Regions Green’s Theorem in the Plane Stoke’s Theorem Divergence Theorem Gauss’ Theorem Variations Stokes’ Theorem Variations Green’s Identities Derivation of Nabla (  ) from Gauss’ Theorem The Operator .
  • 5. 5 Vector AnalysisSOLO TABLE OF CONTENT (continue – 3) Scalar and Vector Fields Fundamental Theorem of Vector Analysis for a Bounded Region V (Helmholtz’s Theorem) Reynolds’ Transport Theorem Poisson’s Non-homogeneous Differential Equation Kirchhoff’s Solution of the Scalar Helmholtz Non-homogeneous Differential Equation Derivation of Nabla (  ) from Gauss’ Theorem The Operator . Orthogonal Curvilinear Coordinates in a Three Dimensional Space Vector Operations in Various Coordinate Systems Applications Laplace Fields Harmonic Functions Rotations
  • 6. 6 Synthetic Geometry Euclid 300BC Algebras HistorySOLO Extensive Algebra Grassmann 1844 Binary Algebra Boole 1854 Complex Algebra Wessel, Gauss 1798 Spin Algebra Pauli, Dirac 1928 Syncopated Algebra Diophantes 250AD Quaternions Hamilton 1843 Tensor Calculus Ricci 1890 Vector Calculus Gibbs 1881 Clifford Algebra Clifford 1878 Differential Forms E. Cartan 1908 First Printing 1482 http://modelingnts.la.asu.edu/html/evolution.html Geometric Algebra and Calculus Hestenes 1966 Matrix Algebra Cayley 1854 Determinants Sylvester 1878 Analytic Geometry Descartes 1637 Table of Content
  • 7. 7 Vector Analysis HistorySOLO John Wallis 1616-1703 1673 Caspar Wessel 1745-1818 “On the Analytic Representation of Direction; an Attempt”, 1799 bia  Jean Robert Argand 1768-1822 1806 1i Quaternions 1843 William Rowan Hamilton 1805-1865 3210 qkqjqiq  Extensive Algebra 1844 Herman Günter Grassmann 1809-1877 “Elements of Vector Analysis” 1881 Josiah Willard Gibbs 1839-1903 Oliver Heaviside 1850-1925 “Electromagnetic Theory” 1893 3. R.S. Elliott, “Electromagnetics”,pp.564-568 http://www-groups.dcs.st-and.ac.uk/~history/index.html Table of Content Edwin Bidwell Wilson 1879-1964 “Vector Analysis” 1901
  • 8. 8 Vector AnalysisSOLO ba   Vector Algebra b  a  a  Addition of Vectors Parallelogram Law of addition Subtraction of Vectors  baba  1 Parallelogram Law of subtraction b  a  b   b  a  ba   Multiplication of Vector by a Scalar am  a  a  am  b  a  ba   ba   b   Geometric Definition of a Vector A Vector is defined by it’s Magnitude and Directiona  a  a 
  • 9. 9 Vector AnalysisSOLO         bababa  ,sin b  a  ba  ,  b  a  ba  ,  ba   abba   Scalar product of two vectors ba  , Vector product of two vectors ba  ,   2/1 2/1 ,cos                   aaaaaaa Magnitude of Vector a  Unit Vector (Vector of Unit Magnitude)aa 1ˆ  a a aa   1 :1:ˆ  Zero Vector (Vector of Zero Magnitude)0  aa  0 0:0   00 a     ababbababa ba   ,cos, ,         Vector Algebra (continue – 1)
  • 10. 10 Vector AnalysisSOLO     n|| ˆˆˆˆ aanannana n    nˆ a   n a n a ˆ ˆ n||      n a n a n ˆ ˆ      Vector decomposition in two orthogonal directions nn ,|| Vector decomposition in two given directions (geometric solution) 1 ˆn a  2 ˆn A B C Given two directions and , and the vector a  1 ˆn 2 ˆn anBCnCA   21 ˆˆ 1 ˆn a  2 ˆn A B Draw lines parallel to those directions passing through both ends A and B of the vector . The vectors obtained are in the desired directions and by rule of vector addition satisfy a  Vector Algebra (continue – 2) Table of Content
  • 11. 11 SOLO Triple Scalar Product Vectors & Tensors in a 3D Space 3321 ,, Eeee   are three non-coplanar vectors, i.e. 1e  2e  3e      0:,, 321321  eeeeee                0,, ,,,, 123123213 132132132321   eeeeeeeee eeeeeeeeeeee   Reciprocal Sets of Vectors The sets of vectors and are called Reciprocal Sets or Systems of Vectors if: 321 ,, eee  321 ,, eee  DeltaKroneckertheis ji ji ee j i j i j i        1 0 Because is orthogonal to and then2e  3e 1 e         321 321321 1 132 1 ,, 1 ,,1 eee keeekeeekeeeeke    and in the same way and are given by:2 e  3 e 1 e             321 213 321 132 321 321 ,,,,,, eee ee e eee ee e eee ee e               
  • 12. 12 SOLO Vectors & Tensors in a 3D Space Reciprocal Sets of Vectors (continue) By using the previous equations we get:                321 3 2 321 13323132 2 321 133221 ,,,,,, eee e eee eeeeeeee eee eeee ee                        321 213 321 132 321 321 ,,,,,, eee ee e eee ee e eee ee e                        0 ,, 1 ,, ,, 321321 3 3321321    eeeeee ee eeeeee    Multiplying (scalar product) this equation by we get:3 e  In the same way we can show that: Therefore are also non-coplanar, and:321 ,, eee     1,,,, 321 321 eeeeee             321 21 3321 13 2321 32 1 ,,,,,, eee ee e eee ee e eee ee e                1e  2e  3e  1 e  2 e  3 e  Table of Content 1e  2e  3e 
  • 13. 13 SOLO Vectors & Tensors in a 3D Space Vector Decomposition Given we want to find the coefficients and such that:3 EA  321 ,, AAA 321 ,, AAA       3 1 3 3 2 2 1 1 3 1 3 3 2 2 1 1 j j j i i i eAeAeAeA eAeAeAeAA   3,2,1, iee i i  are two reciprocal vector bases Let multiply the first row of the decomposition by :j e  Let multiply the second row of the decomposition by :ie  j i j i i i j i ij AAeeAeA    3 1 3 1   i j i j j j i j ji AAeeAeA    3 1 3 1   Therefore: ii jj eAAeAA   & Then:                      3 1 3 3 2 2 1 1 3 1 3 3 2 2 1 1 j j j i i i eeAeeAeeAeeA eeAeeAeeAeeAA   Table of Content 1e  2e  3e 
  • 14. 14 SOLO Vectors & Tensors in a 3D Space The Summation Convention j j j j j eAeAeAeAeA    3 1 3 3 2 2 1 1 The last notation is called the summation convention, j is called the dummy index or the umbral index.             i i i i j j j j j j j j j i i i i i eAeeAeeAeeA eAeeAeeAeeAA         3 1 3 1 Instead of summation notation we shall use the shorter notation first adopted by Einstein  3 1j j j eA  j j eA  Table of Content
  • 15. 15 SOLO Vectors & Tensors in a 3D Space Let define: The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee   jiijjiij geeeeg   3321 ,, Eeee   the metric covariant tensors of By choosing we get:         j ijiii j jiiiii egegegeg eeeeeeeeeeeee     3 3 2 2 1 1 3 3 2 2 1 1 i eA   or: j iji ege   For i = 1, 2, 3 we have:                                                        3 2 1 332313 322212 312111 3 2 1 333231 232221 131211 3 2 1 e e e eeeeee eeeeee eeeeee e e e ggg ggg ggg e e e             1e  2e  3e 
  • 16. 16 SOLO Vectors & Tensors in a 3D Space We want to prove that the following determinant (g) is nonzero: The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee                            332313 322212 312111 333231 232221 131211 detdet: eeeeee eeeeee eeeeee ggg ggg ggg g    g is the Gram determinant of the vectors 321 ,, eee  Jorgen Gram 1850 - 1916 Proof: Because the vectors are non-coplanars the following equations:321 ,, eee  03 3 2 2 1 1  eee   is true if and only if 0321   Let multiply (scalar product) this equation, consecutively, by :321 ,, eee                                             0 0 0 0 0 0 3 2 1 332313 322212 312111 33 3 23 2 13 1 32 3 22 2 12 1 31 3 21 2 11 1       eeeeee eeeeee eeeeee eeeeee eeeeee eeeeee       Therefore α1= α2= α3=0 if and only if g:=det {gij}≠0 q.e.d.
  • 17. 17 SOLO Vectors & Tensors in a 3D Space Because g ≠ 0 we can take the inverse of gij and obtain: The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee   where Gij = minor gij having the following property: j i kj ik ggG                                                      3 2 1 333231 232221 131211 3 2 1 333231 232221 131211 3 2 1 1 e e e ggg ggg ggg e e e GGG GGG GGG g e e e          and: g G g gminor g ij ijij  Therefore: g g g g G gg j i kj ik kj ik   j i kj ik gg  Let multiply the equation by gij and perform the summation on ij iji ege   jj ij ij i ij eeggeg   Therefore: i ijj ege   Let multiply the equation byk kjj ege   i e  iji k kjijji geegeeee   jiijjiij geeeeg   i jkj ik ggG  The Operator .
  • 18. 18 SOLO Vectors & Tensors in a 3D Space The Metric Tensor or Fundamental Tensor Specified by .3321 ,, Eeee   Let find the relation between g and    321321 :,, eeeeee   We shall write the decomposition of in the vector base32 ee   321 ,, eee  3 3 2 2 1 1 32 eeeee    Let find λ1, λ2, λ3. Multiply the previous equation (scalar product) by .1e      i i ggggeeeeee 113 3 12 2 11 1 321321 ,,    Multiply this equation by g1i:   ii i ii ggeeeg     1 1 1321 1 ,, Therefore:  321 1 ,, eeeg ii   Let compute now:            321 1 0 323 3 0 322 2 321 1 3232 eeeeeeeeeeeeeeee                                321 11 321 11 321 2 233322 321 3322332 321 3232 321 32321 ,,,,,,,, ,,,, eee gg eee G eee ggg eee eeeeeee eee eeee eee eeee                From those equations we obtain:  321 11 1 ,, eee gg  Finally:     geeeeee  321 2 321 1 ,,,,   We can see that if are collinear than and g are zero.321 ,, eee  321 ,, eee  Table of Content
  • 19. 19 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation Let choose another base and its reciprocal 321 ,, fff   321 ,, fff                                                 3 2 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 e e e L e e e f f f ef j j ii               where j i j i ef   By tacking the inverse of those equations we obtain:                                                   3 2 1 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 f f f L f f f e e e fe i j ij               where j ij i ef   Because are the coefficients of the inverse matrix with coefficients :j i  j i i j i k k j  
  • 20. 20 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation (continue – 1) Let write any vector in those two bases:A                                                       3 2 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 e e e L e e e f f f ef ef ji j i j j ii                 then:                                                     3 2 1 1 3 2 1 3 3 3 2 3 1 2 3 2 2 2 1 1 3 1 2 1 1 3 2 1 E E E L E E E F F F ef EF T j ii j i j ji       i i j j fFeEA   iijj fAFeAE   & i j ji i i j j j j i i EFfEeEfF    or: But we remember that: We can see that the relation between the components F1, F2, F3 to E1, E2, E3 is not similar, contravariant, to the relation between the two bases of vectors to . Therefore we define F1, F2, F3 and E1, E2, E3 as the contravariant components of the bases and .  321 ,, fff   321 ,, eee   321 ,, fff   321 ,, eee  where
  • 21. 21 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation (continue – 2)                                              3 2 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 E E E L E E E F F F EF j iji     i i j j fFeEA   iijj fAFeAE   & then: Let write now the vector in the two bases andA   321 ,, fff   321 ,, eee  where and j ij ef ijjjjii EfeEeEAfAF j iji      We can see that the relation between the components F1, F2, F3 to E1, E2, E3 is similar, covariant, to the relation between the two bases of vectors to . Therefore wew define F1, F2, F3 and E1, E2, E3 as the covariant components of the bases and .  321 ,, fff   321 ,, eee   321 ,, fff   321 ,, eee 
  • 22. 22 SOLO Vectors & Tensors in a 3D Space We have: Change of Vector Base, Coordinate Transformation (continue – 3)     j j j j i i i i eeAeA eeAeAA     Ai contravariant component Aj covariant component Let find the relation between covariant and the contravariant components: j j j ij i ege i i eAegAeAA j iji     i i i ij j ege j j eAegAeAA i ijj     Therefore: ij j i ij i j gAAgAA  & Let find the relation between gij and gij defined in the bases and to and defined in the bases and . ie  i e  i f  i f  ij g ij g m m kkj j ii efef    & jm m k j imj m k j ikiik geeffg    Hence: jm m k j iik gg  This is a covariant relation of rank two, (similar, two times, to relation between to .i f  j e 
  • 23. 23 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation (continue – 4)                                                      3 2 1 3 2 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 3 2 1 e e e L e e e f f f ef ef ji j i j j ii                 Since we have:k iki fgf     m jm j i ege j j i ef k iki egefgf m jmjj j ii       and:     m jm j i km kjm j i gg k ik egfgfg mjm m k j iik     Therefore, by equalizing the terms that multiply we obtain:                                                  3 2 1 3 2 1 3 3 3 2 3 1 2 3 2 2 2 1 1 3 1 2 1 1 3 2 1 f f f L f f f e e e fe Tkm k m                jm j i g We found the relation:
  • 24. 24 SOLO Vectors & Tensors in a 3D Space Change of Vector Base, Coordinate Transformation (continue – 5) Therefore:                                                 3 2 1 1 3 2 1 3 3 3 2 3 1 2 3 2 2 2 1 1 3 1 2 1 1 3 2 1 e e e L e e e f f f ef Tmj m j               Let take the inverse of the relation by multiplying by and summarize on m:j m km k m fe   jkj k km k j m mj m fffe    From the relation: mk m kjj i i efef    & we have: jmk m j i mjk m j i kiik geeffg    or: jmk m j i ik gg  This is a contravariant relation of rank two. From the relation: m m kk jj i i efef    & we have: m j m k i jm jm k i j i kk i eeff    or: This is a relation once covariant and once contravariant of rank two. m j m k i j i k   Table of Content
  • 25. 25 SOLO Vectors & Tensors in a 3D Space Cartesian Coordinates Three dimensional cartesian coordinates are define as coordinates in a orthonormal basis such that:     zyxorkjieee 1,1,1ˆ,ˆ,ˆ,, 321   ix ˆ1  jy ˆ1  O kz ˆ1  x1 y1z1 0111111 1111111   zyzxyx zzyyxx yzxxyzzxy yxzxzyzyx 111111111 111111111       11111,1,12  zyxzyxg The reciprocal set is identical to the original set       ji ji ggg ij ijj iij 0 1  Given              z y x zyx A A A zyxzAyAxAA 111111               z y x A A A AMatrix notation of a vector
  • 26. 26 SOLO Vectors & Tensors in a 3D Space Cartesian Coordinates (continue – 1) ix ˆ1  jy ˆ1  O kz ˆ1  Given              z y x zyx A A A zyxzAyAxAA 111111               z y x A A A AMatrix notation of a vector              z y x zyx B B B zyxzByBxBB 111111                ABABBABABABA B B B AAAzByBxBzAyAxABA T zzyyxx T z y x zyxzyxzyx               111111              z y x B B B B
  • 27. 27 SOLO Vectors & Tensors in a 3D Space Cartesian Coordinates (continue – 2)           zyx zyxxyyxzxxzyzzy zyxzyx BBB AAA zyx zBABAyBABAxBABA zByBxBzAyAxABA 111 det111 111111             ABAB A A A BB BB BB BA B B B AA AA AA BA z y x xy xz yz z y x xy xz yz                                                   0 0 0 0 0 0 ix ˆ1  jy ˆ1  O kz ˆ1  Given              z y x zyx A A A zyxzAyAxAA 111111               z y x A A A AMatrix notation of a vector              z y x zyx B B B zyxzByBxBB 111111               z y x B B B B
  • 28. 28 SOLO Vectors & Tensors in a 3D Space Cartesian Coordinates (continue – 2) Table of Content                           ACBACBCBACBCBACBCB BACBACACBACACBACAC CBABACBABACBABA CCC BBB AAA BBB AAA CCC zCyCxC BBB AAA zyx CBA zxyyxyzxxzxyzzy zxyyxyzxxzxyzzy zxyyxyzxxzxyzzy zyx zyx zyx zyx zyx zyx zyx zyx zyx        detdet111 111 det                  ABCABCBAC B B B AA AA AA CCCBAC TT z y x xy xz yz zyx                           0 0 0 Given              z y x zyx A A A zyxzAyAxAA 111111               z y x A A A A Matrix notation of a vector              z y x zyx B B B zyxzByBxBB 111111               z y x B B B B              z y x zyx C C C zyxzCyCxCC 111111               z y x C C C C
  • 29. 29 Vector AnalysisSOLO            bacbacacbacbcbacba  ,,,,:,,        cbabcacba                    cbdadbca dcbcdba dcbadcba                  adcbbdca dcbacdbadcba   ,,,, ,,,,          2 ,, cbaaccbba                          feabdcfebadc dcefbadcfeba fedcbafecdbafedcba    ,,,,,,,, ,,,,,,,, ,,,,,,,,    Vector Identities Summary       0 bacacbcba  Table of Content
  • 30. 30 SOLO VECTOR SPACE Given the complex numbers .C ,, A Vector Space V (Linear Affine Space) with elements over C if its elements satisfy the following conditions: Vzyx   ,, I. Exists a operation of Addition with the following properties: xyyx   Commutative (Abelian) Law for Addition1    zyxzyx   Associative Law for Addition2 xx   0 Exists a unique vector0  3 II. Exists a operation of Multiplication by a Scalar with the following properties: 0..   yxtsVyVx4 Inverse xx  15    xx    Associative Law for Multiplication6   xxx    Distributive Law for Multiplication7   yxyx    Commutative Law for Multiplication8           00101010 3 575   xxxxxxxxWe can write: Vector Analysis
  • 31. 31 SOLO Scalar Product in a Vector Space The Scalar Product of two vectors is the operation with the symbol with the following properties: Vyx   ,   Cyx   ,    xyyx  ,,     yxyx  ,,        zyzxzyx  ,,,      00,&0,   xxxxx Distance Between Two Vectors The Distance between two Vectors is defined by the following properties: Vyx   ,   yxyxd  ,     00,&0,   xxxdxxd    xydyxd  ,,       yzdzxdyxd  ,,,  Vector Analysis Table of Content
  • 32. 32 SOLO Differential Geometry is the study of geometric figures using the methods of Calculus. Here we present the curves and surfaces embedded in a three dimensional space. Properties of curves and surfaces which depend only upon points close to a particular point of the figure are called local properties.. The study of local properties is called differential geometry in the small. Those properties which involve the entire geometric figure are called global properties. The study of global properties is called differential geometry in the large. Hyperboloid of RotationToroyd Mobius Movement Differential Geometry Differential Geometry in the 3D Euclidean Space Table of Content
  • 33. 33 SOLO Differential Geometry in the 3D Euclidean Space A curve C in a three dimensional space is defined by one parameter t,  tr   ur  rd P O a b C Theory of Curves Regular Parametric Representation of a Vector Function: parameter t, defined in the interval I and:   Ittrr  ,   tr  (i) is of class C1 (continuous and 1st order differentiable) in I Arc length differential:      td td rd td td rd td rd trdtrdsd          2/1 2/1 : We also can define      sdtrdtrdsd  2/1* :  (ii)   It td trd  0    Iinconstantnottr   Arc length as a parameter:  t t td td rd s 0  Regular Curves: A real valued function t = t (θ), on an interval Iθ, is an allowable change of parameter if: (i) t (θ) is of class C1 in Iθ (ii) d t/ d θ ≠ 0 for all θ in Iθ A representation on Is is a representation in terms of arc length or a natural representation  srr   Table of Content
  • 34. 34 SOLO Differential Geometry in the 3D Euclidean Space A curve C in a three dimensional space is defined by one parameter t,  tr   ur  rd P O a b C - arc length differential     td td rd td rd trdtrdsd 2/1 2/1 :          td rd td rd r sd rd t /::   - unit tangent vector of path C at P (tangent to C at P) 1x 2x 3x td rd r '  - tangent vector of path C at P (tangent to C at P) 0,0,sincos 321  baetbetaetar  Example: Circular Helix 0,0,cossin' 321  baebetaeta td rd r     2/122 2/1 ba td rd td rd td rd            321 2/122 cossin/: ebetaetaba td rd td rd t   Theory of Curves (continue – 1) We also can define      sdtrdtrdsd  2/1* :  t sd rd sd rd  * Unit Tangent Vector of path C at a point P Table of Content
  • 35. 35 SOLO Differential Geometry in the 3D Euclidean Space The earliest investigations by means of analysis were made by René Descartes in 1637.  tr   ur  rd P O a b C René Descartes 1596 - 1650 Pierre Fermat 1601 - 1665 Christian Huyghens 1629 - 1695 Gottfried Leibniz 1646 - 1716 The general concept of tangent was introduced in seventeenth century, in connexion with the basic concepts of calculus. Fermat, Descartes and Huyghens made important contributions to the tangent problem, and a complete solution was given by Leibniz in 1677. The first analytical representation of a tangent was given by Monge in 1785. Gaspard Monge 1746 - 1818 Theory of Curves (continue – 2)
  • 36. 36 SOLO Differential Geometry in the 3D Euclidean Space A curve C in a three dimensional space is defined by one parameter t,  tr  - arc length differential     td td rd td rd trdtrdsd 2/1 2/1 :          '/'/:: rr td rd td rd r sd rd t   - unit tangent vector of path C at P (tangent to C at P) Normal Plane to at P:t    00  trr  We also can define - arc length differential     sdtrdtrdsd  2/1* :  t sd rd sd rd  * O a C t  P r  b 0 r  NormalPlane   00  trr  Theory of Curves (continue – 3) Return to Table of Contents
  • 37. 37 SOLO Differential Geometry in the 3D Euclidean Space O a C t  P r  b NormalPlane   00  trr  0 r  Curvature of curve C at P: rt sd td k   : Since 01  tkttt sd td tt    Define nnkkkkkkn    1 /1:&/:  ρ – radius of curvature of C at P k – curvature of C at P A point on C where k = 0 is called a point of inflection and the radius of curvature ρ is infinite. '' st td sd sd rd td rd r           "'"'"' ' ''''' 22 stskstststs td sd sd td td sd ts td td st td d r td d r           32 '"''''' skntstskstrr   '' sr   3 1 '''' skntrr    3 ' ''' r rr k     Let compute k as a function of and :'r  ''r  Theory of Curves (continue – 4)
  • 38. 38 SOLO Differential Geometry in the 3D Euclidean Space 1x 2x 3x t  k  0,0,sincos 321  baetbetaetar  Example 2: Circular Helix 0,0,cossin' 321  baebetaeta td rd r       2/1222/122 2/1 bardsdba td rd td rd td rd             321 2/122 cossin/: ebetaetaba td rd td rd t    2122 sincos/ etet ba a td sd td rd t sd td k       1 x 2x 3 x t  k  0,sincos 21  aetaetar  Example 1: Circular Curve 0,cossin' 21  aetaeta td rd r       2/1222/122 2/1 bardsdba td rd td rd td rd           21 cossin/: etaetaa td rd td rd t    21 sincos 1 / etet atd sd td rd t sd td k     Theory of Curves (continue – 5) Table of Content
  • 39. 39 SOLO Differential Geometry in the 3D Euclidean Space O a C t  P b ntk   1  NormalPlane Osculating Plane   00  trr  0r    00  ktrr Osculating Plane of C at P is the plane that contains and P:     00  ktrr kt  , The name “osculating plane” was introduced by D’Amondans Charles de Tinseau (1748-1822) in 1780. O a C t  P b ntk   1  NormalPlane Osculating Plane   00  trr  0r    00  ktrr  The osculating plane can be also defined as the limiting position of a plane passing through three neighboring points on the curve as the points approach the given point. If the curvature k is zero along a curve C then: tarrconstartt   0 0 The curve C is a straight line. Conversely if C is a straight line: 0//0  tkaa td rd td rd ttarr  C a regular curve of class ≥2 (Cclass) is a straight line if and only if k = 0 on C Theory of Curves (continue – 6) Table of Content
  • 40. 40 SOLO Differential Geometry in the 3D Euclidean Space Osculating Circleof C at P is the plane that contains and Pkt  , Theory of Curves (continue – 6) The osculating circle of a curve C at a given point P is the circle that has the same tangent as C at point P as well as the same curvature. Just as the tangent line is the line best approximating a curve at a point P, the osculating circle is the best circle that approximates the curve at P. http://mathworld.wolfram.com/OsculatingCircle.html O a C t  P b ntk   1  Normal Plane Osculating Plane   00  trr  0r    00  ktrr  Osculating Circle Osculating Circles on the Deltoid The word "osculate" means "to kiss."
  • 41. 41 SOLO Differential Geometry in the 3D Euclidean Space Osculating Circleof C at P is the plane that contains and P kt  , Theory of Curves (continue – 6a) O a C t  P b ntk   1  Normal Plane Osculating Plane   00  trr  0r    00  ktrr  Osculating Circle 3 xy  xy /1 xy cos xy sin http://curvebank.calstatela.edu/osculating/osculating.htm xy tan Table of Content
  • 42. 42 SOLO Differential Geometry in the 3D Euclidean Space O a C t  P b ntk   1  NormalPlane Osculating Plane   00  trr  0r    00  ktrr b  Rectifying Plane   00  krr  Binormal ntb  : Tangent Line: Principal Normal Line: Binormal Line: Normal Plane: Rectifying Plane: Osculating Plane: tmrr   0 nmrr   0 bmrr   0   00  trr    00  nrr    00  brr  The name binormal was introduced by Saint-Venant Jean Claude Saint-Venant 1797 - 1886 Fundamental Planes:Fundamental Lines: Theory of Curves (continue – 7) Table of Content
  • 43. 43 SOLO Differential Geometry in the 3D Euclidean Space Torsion Suppose that is a regular curve of class ≥ 3 (Cclass) along which is of class C1. then let differentiate to obtain:  srr    sn       snstsb                        snstsnstsnsnksnstsnstsb   Since                 001  snsnsnsnsnsnsnsn  Therefore is normal to , meaning that is in the rectifying plane, or that is a linear combination of and . n n t  b           sbsstssn                        snssbsstsstsnstsb    O a C t  P b n  0r  b  The continuous function τ (s) is called the second curvature or torsion of C at P.      snsbs   Theory of Curves (continue – 8)
  • 44. 44 SOLO Differential Geometry in the 3D Euclidean Space Torsion (continue – 1) Suppose that the torsion vanishes identically (τ ≡0) along a curve , then srr           0 0 bsbsnssb    O a C t  P b n  0r  0 b  Since and are orthogonal st   sb          constbsrbtbsr sd d bsr sd d  0000 0  Therefore is a planar curve confined to the plane srr     constbsr  0  C a regular curve of class ≥3 (Cclass) is a planar curve if and only if τ = 0 on C 1x 2x 3x t  k  0,0,sincos 321  baetbetaetar  Example 2: Circular Helix    321 2/122 cossin ebetaetabat    21 sincos etetn            321 2/122 21321 2/122 cossin sincoscossin eaetbetbba etetebetaetabantb          21 1222/122 sincos etbetbbaba td bd sd td td bd sd bd b      122   babnb   Theory of Curves (continue – 9)
  • 45. 45 SOLO Differential Geometry in the 3D Euclidean Space Torsion (continue – 2) Let compute τ as a function of and :'',' rr  '''r  ttr sd td td rd sd rd r      '      tbknkttrtrtr sd d trtr sd d r    2 "''''       tkbkbknkbktnkbktbktbkbk trttrtrtrttrttrtrtrtr sd d r      2 332 '''"3''''"2"'"'                                        2 0 3 1 2 0 26 6 0 3 0 4 0 22 5232 32 ,,,,,,'''",' '''",'',",'''',','",','3 '''"'"''''"'3' '''"3'"'',, ktntkbntknntkktkbknknktrrrt rrrtrrrttrrrttrrrtt rrtrrttrrttrrtttr trttrtrtrtrtrrrr                    ' 1 / 1 rtdsdsd td t   3 ' ''' r rr k    We also found:      6 2 2 6 ' ''' ' '''",' ,, r rr k r rrr rrr            2 ''' '''",' rr rrr      Theory of Curves (continue – 10) Table of Content
  • 46. 46 SOLO Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations Theory of Curves (continue – 11) We found and      snssb       snskst   Let differentiate      stsbsn                                stsksbssnsbskstsnsstsbstsbsn     We obtain          sbsnskstst  00             sbsbsnstsksn   0          sbsnsstsb  00   or                                                        sb sn st s ssk sk sb sn st       00 0 00   Jean Frédéric Frenet 1816 - 1900 Those are the Serret – Frenet Equations of a curve. Joseph Alfred Serret 1819 - 1885
  • 47. 47 SOLO Let compute: Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations (continue – 1) Theory of Curves (continue – 12) Let show that if two curves C and C* have the same curvature k (s) = k* (s) and torsion τ (s) = τ*(s) for all s then C and C* are the same except for they position in space. Assume that at some s0 the triads and coincide.      999 ,, sbsnst       999 *,*,* sbsnst     ******** * nttnknkttnktttttt sd d kk             ************ * * bnnbnttnkbtknnbtknnnnnn sd d kk          ******** * nbbnnbbnbbbbbb sd d      Adding the equations, we obtain:   0***  bbnntt sd d  Integrating we obtain:     30 ******  sbbnnttconstbbnntt  Since: and1,,1 ***  bbnntt    3***  bbnntt  we obtain: 1***  bbnntt  Finally since:         constsrsr sd rd stst sd rd  * * *    
  • 48. 48 SOLO Existence Theorem for Curves Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations (continue – 2) Theory of Curves (continue – 13) Let k (s) and τ (s) be continuous functions of a real variable s for s0 ≤ s ≤ sf. Then there exists a curve , s0 ≤ s ≤ sf, of class C2 for which k is the curvature, τ is the torsion and s is a natural parameter.  srr   332211332211332211 ,, ebebebbenenennetetett          tnktttttt sd d   2          nbntknnnnnn sd d   22    bnbbbbbb sd d               2 with: Proof: Consider the system of nine scalar differential equations:                       3,2,1,,,  isnssbsbsstsksnsnskst iiiiiii   and initial conditions:       302010 ,, esbesnest              btttktnkntntnt sd d             nnbbbtkbnbnbn sd d                  ntbnkbtbtbt sd d          and initial conditions:             1,0,1,0,0,1 000000  ssssss bbbnnnbtnttt 
  • 49. 49 SOLO Existence Theorem for Curves (continue – 1) Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations (continue – 3) Theory of Curves (continue – 14)                                    bnbb sd d ntbnkbt sd d nnbbbtkbn sd d btttktnknt sd d nbntknn sd d tnktt sd d          2 222 Proof (continue – 1): and initial conditions:             1,0,1,0,0,1 000000  ssssss bbbnnnbtnttt  We obtain: The solution of this type of differential equations with given initial conditions has a unique solution and since is a solution, it is unique.             1,0,1,0,0,1  bbbnnnbtnttt  The solution is an orthonormal triad.bnt  ,, We now define the curve:     s s dtsrr 0 :   We have: and , therefore k (s) is the curvature.1 tr          1&  snsnskst  Finally since:       nbtttknnkntntbntb    Therefore τ (s) is the torsion of  srr   q.e.d.
  • 50. 50 SOLO From the previous development we can state the following theorems: Differential Geometry in the 3D Euclidean Space Seret-Frenet Equations (continue – 4) Theory of Curves (continue – 15) A curve is defined uniquely by the curvature and torsion as functions of a natural parameter. The equations k = k (s), τ = τ (s), which give the curvature and torsion of a curve as functions of s are called the natural or intrinsec equations of a curve, for they completely define the curve. O 0s C t  P n  0r  b  k 1  f s Fundamental Existence and Uniqueness Theorem of Space Curves Let k (s) and τ (s) be arbitrary continuous functions on s0≤s≤sf. Then there exists, for position in space, one and only one space curve C for which k (s) is the curvature, τ (s) is the torsion and s is a natural parameter along C. O 0s C t  P n  b  f s * C 0 r  * 0r  Table of Content
  • 51. 51 SOLO Let consider a space curve C. We construct the tangent lines to every point on C and define an involute Ci as any curve which is normal to every tangent of C. Differential Geometry in the 3D Euclidean Space Involute Theory of Curves (continue – 16) From the Figure we can see that the equation of the Involute is given by: turr  1 Differentiating this equation we obtain: 11 1 1 1 sd sd t sd ud nkut sd sd t sd ud sd td u sd rd t sd rd                  Scalar multiply this equation by and use the fact that and from the definition of involute : t  0nt  01 tt     1101 10 sd sd tt sd ud ntkutttt         01  sd ud scu         stscsrsr  1 C i C O r  1r  t  1 t  s c  Involute Curve
  • 52. 52 SOLO Differential Geometry in the 3D Euclidean Space Involute (continue – 1) Theory of Curves (continue – 17) C i C O r  1r  t  1 t  s c  Involute Curve        stscsrsr  1      n sd sd ksc sd sd t sd td sc sd rd sd rd t t     111 1 1             and are collinear unit vectors, therefore:1t  n     kscsd sd sd sd ksc   1 1 11 The curvature of the involute, k1, is obtained from:      ksc btk kscsd nd sd sd sd td nk sd td nt kscsd sd             11 1 1 1 1 11 1 1 Hence:   22 22 2 1 ksc k k     For a planar curve (τ=0) we have:   t sc nk     1 011 
  • 53. 53 SOLO Differential Geometry in the 3D Euclidean Space Involute (continue – 3) Theory of Curves (continue – 18) C i C O r  1r  t  1 t  s c  Involute Curve http://mathworld.wolfram.com/Involute.html Table of Content
  • 54. 54 SOLO The curve Ce whose tangents are perpendicular to a given curve C is called the evolute of the curve. Differential Geometry in the 3D Euclidean Space Evolute Theory of Curves (continue – 19) 11 twrbvnurr   Differentiating this equation we obtain:   11 1 1 1 sd sd b sd vd n sd ud nvbtkut sd sd b sd vd n sd ud sd bd v sd nd u sd rd t sd rd                      Scalar multiply this equation by and use the fact that and from the definition of evolute : t  0 btnt  01 tt    111 1 0 sd sd ttkutttt         01  ku  k u 1 C e C O r  1r  t 1 t  Evolute Curve The tangent to Ce, , must lie in the plane of and since it is perpendicular to . Therefore: n  b  t 1t  1 1 sd sd n sd ud vb sd vd ut                       
  • 55. 55 SOLO Differential Geometry in the 3D Euclidean Space Evolute (continue – 1) Theory of Curves (continue – 20)    ccuv   tantan  k u 1 C e C O r  1r  t 1 t  Evolute Curve We obtained: 1 1 sd sd n sd ud vb sd vd ut                            111 // wbvnuwrrt  But: Therefore: v v sd ud u u sd vd     or:                  u v sd d vu sd ud v sd vd u 1 22 tan c u v ds s s          1 tan 0  and:  bcnrr    tan1 We have one parameter family that describes the evolutes to the curve C.
  • 56. 56 SOLO Differential Geometry in the 3D Euclidean Space Evolute (continue – 2) Theory of Curves (continue – 21) http://math.la.asu.edu/~rich/MAT272/evolute/ellipselute.html Evolute of Ellipse Evolute of Logarithmic Spiral also a Logarithmic Spiral Evolute of Parabola Table of Content C e C O r  1r  t 1 t  Evolute Curve
  • 57. 57 SOLO Differential Geometry in the 3D Euclidean Space The vector defines a surface in E3  vur ,  vu vu rr rr N       vur ,   vdvudur  ,  rd 2 rd r udru  vdrv  d Nd P O  vudur ,               22 2 22 22 2 2 ,2 2 1 , 2 1 ,,, vdudOvdrvdudrudrvdrudr vdudOrdrdvurvdvudurvur vvvuuuvu     The vectors and define the tangent plane to the surface at point P. P u u r r      P v v r r      Define: Unit Normal Vector to the surface at P vu vu rr rr N     : First Fundamental Form:       2222 22: vdGvdudFudEvdrrvdudrrudrrrdrdI vvvuuu         0 2 0,0,00:                  GF FEforConditionSylvester FEGGE vd ud GF FE vdudrdrdI Surfaces in the Three Dimensional Spaces Table of Contents
  • 58. 58 SOLO Arc Length on a Path on the Surface:                 b a b a vuvu b a tdvdrudrvdrudrtd td rd td rd td td rd L 2/1 2/1                                                                           b a b a td td vd td ud GF FE td vd td ud td td vd G td vd td ud F td ud EL 2/1 2/1 22 2 Surface Area:  vur ,  rd udru  vdrv  d P O      vdudFGEvdud GE F GE vdud rr rr rrvdudrrrr vdudrrrrvdudrrvdrudrd vu vu vuvuvu vuvuvuvu 2 2/1 2 2/1 2 2/12 1 1,cos1 ,sin                                 vdudFGEd 2  vur ,  rd udru  vdrv P O a b Differential Geometry in the 3D Euclidean Space Table of Contents
  • 59. 59 SOLO Change of Coordinates  vur ,  rd udru  vdrv  d P O vdrv  udru  vdrudrvdrudrd vuvu   vdudFGEvdud vu vu JFGEvdudFGEd 222 , ,           vurvurr ,,  Change of coordinates from u,v to θ,φ    vuvv vuuu , ,  The coordinates are related by                   v u vv uu vd ud vu vu       I vd ud GF FE vdud vd ud vv uu GF FE vu vu vdud vd ud GF FE vdudI vu vu vv uu                                                      td td vd td ud GF FE td vd td ud td td vd td ud GF FE td vd td ud td td rd td rd Ld 2/12/1 2/1                                                                                                                         vu vu JFGE vv uu FGE vv uu GF FE vu vu GF FE FGE vu vu vu vu vv uu , , detdetdetdetdet 22 ** ** 2 Arc Length on a Path on the Surface and Surface Area are Invariant of the Coordinates: First Fundamental Form is Invariant to Coordinate Transformation Differential Geometry in the 3D Euclidean Space Table of Contents
  • 60. 60 SOLO vu vu rr rr N       vur ,   vdvudur  ,  rd 2 rd r udru  vdrv  d Nd P O  vudur ,  Second Fundamental Form: NdrdII :           22 2 2 2 2 : vdNvdudMudL vdNrvdudNrNrudNr vdNudNvdrudrNdrdII N vv M uvvu L uu vuvu             vdNudNNdNNdNN vu   01                NrNrNrNrNr vd d NrNrNrNrNr ud d Nr vuvuvuvuu uuuuuuuuu u    0 0 0                NrNrNrNrNr vd d NrNrNrNrNr ud d Nr vvvvvvvvv vuuvuvvuv v    0 0 0 Differential Geometry in the 3D Euclidean Space
  • 61. 61 SOLO vu vu rr rr N       vur ,   vdvudur  ,  rd 2 rd r udru  vdrv  d Nd P O  vudur ,  Second Fundamental Form: NdrdII :       2 2 2 : vdNrvdudNrNrudNrNdrdII N vv M uvvu L uu          NrNr uuuu   NrNr vuuv         NrNr NrL uuuu uu            uvvu vuuv vuvu NrNrM NrNr NrNr          NrNr NrN vvvv vv   NrNr vuvu   NrNr vvvv   22 2: vdNvdudMudLNdrdII  NrL uu   NrM vu   NrN vv   Differential Geometry in the 3D Euclidean Space
  • 62. 62 SOLO vu vu rr rr N       vur ,  O  vdvudur  , udru  vdrv  rd Second Fundamental Form: NdrdII :                33 3 3223 22 33 3 32 ,33 6 1 2 2 1 , 6 1 2 1 ,,, vdudOvdrvdudrvdudrudr vdrvdudrudrvdrudr vdudOrdrdrdvurvdvudurvur vvvvuvvuuuuu vvvuuuvu                 IINvdudOvdNvdudMudL NvdudOvdNrvdudNrudNr NvdudONrdNrdNrdNr vvvuuu 2 1 ,2 2 1 ,2 2 1 , 6 1 2 1 22 2 22 22 2 22 33 3 32 0         Differential Geometry in the 3D Euclidean Space
  • 63. 63 SOLO N  Second Fundamental Form: NdrdII : N  N  (i) Elliptic Case (ii) Hyperbolic Case (iii) Parabolic Case 02 MNL 02 MNL 0 &0 222 2   MNL MNL Differential Geometry in the 3D Euclidean Space
  • 64. 64 SOLO Differential Geometry in the 3D Euclidean Space (continue – 6a)  vur ,  vdrv  P O N  1nr  2nr  udru  2 M 1 M 02  MNL Dupin’s Indicatrix N  1n r  2n r  P 2 M 1 M 02  MNL N  1nr 2nr  P 1M 2M 0 0 222 2   MNL MNL http://www.mathcurve.com/surfaces/inicatrixdedupin/indicatrixdedupin.html Pierre Charles François Dupin 1784 - 1873 We want to investigate the curvature propertiesat a point P.     IINvdudOvdNvdudMudLNr 2 1 ,2 2 1 22 2 22   The expression 12 2 221 2 1  xNxxMxL was introduced by Charles Dupin in 1813 in “Développments de géométrie”, to describe the local properties of a surface. Second Fundamental Form: NdrdII : http://www.groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html Differential Geometry in the 3D Euclidean Space
  • 65. 65 SOLO N  Second Fundamental Form: NdrdII : N  (iv) Planar Case 0 MNL      3223 33 3 3223 6 1 ,33 6 1 vdDvdudCvdudBudA vdudOvdrNvdudrNvdudrNudrNNr vvvvuvvuuuuu    DxCxBxA  23 has 3 real roots Monkey Saddle DxCxBxA  23 has one real root Differential Geometry in the 3D Euclidean Space
  • 66. 66 SOLO Second Fundamental Form: NdrdII :    vurvurr ,,  Change of coordinates from u,v to θ,φ    vuvv vuuu , ,  The coordinates are related by                   v u vv uu vd ud vu vu          2222 22 uuuuuvvuuvuuuuuu vNvuMuLNvrvururNrL       vuvuvuvuvuvvvuvuvuuvvuuuvu vvNvuuvMuuLNvvrvuruvruurNrM            2222 2 vvvvvvvvvuvvvvuvuuvv vNvuMuLNvruvrvururNrN   Unit Normal Vector to the surface at P vu vu vu vu rr rr rr rr N          : uvuuvuu vrur u v r u u rr         vvvuvuv vrur v v r v u rr               II vd ud NM ML vdud vd ud vv uu NM ML vu vu vdud vd ud NM ML vdudII vu vu vv uu                                                      Second Fundamental Form is Invariant (unless the sign) to Coordinate Transformation Differential Geometry in the 3D Euclidean Space Table of Contents
  • 67. 67 SOLO N  Osculating Plane of C at P Principal Normal Line of C at P Surface t  P k  n1  vur ,  Normal Curvature - Length differential  2/1 rdrdrdsd      tvturr ,  Given a path on a surface of class Ck ( k ≥ 2) we define: td rd td rd sd rd t /:   - unit vector of path C at P (tangent to C at P) td rd td td sd td k /:   - curvature vector of path C at P       curvatureofradius nn nnk sd td k  111 1 1 1   NNkkn  : - normal curvature vector to C at P      /coscos1 :   kNnk Nkkn   - normal curvature to C at P Differential Geometry in the 3D Euclidean Space
  • 68. 68 SOLO N  Osculating Plane of C at P Principal Normal Line of C at P Surface t  P k  n1  vur ,  Normal Curvature (continue – 1) N Because C is on the surface, is on the tangent plan normal to . t    td Nd tN td td td Nd tN td td Nt td d Nt        00 and         vdrudrvdrudrvdNudNvdrudr td rd td rd td Nd td rd td rd td Nd td rd td rd td Nd t td rd N td td N sd td Nkk vuvuvuvu n                   / / /// 2 G vd ud F vd ud E N vd ud M vd ud L I II vdGvdudFudE vdNvdudMudL td vd G td vd td ud F td ud E td vd N td vd td ud M td ud L kn                                                                              2 2 2 2 2 2 2 2 22 22 22 22 Differential Geometry in the 3D Euclidean Space
  • 69. 69 SOLO Normal Curvature (continue – 2) G vd ud F vd ud E N vd ud M vd ud L I II vdGvdudFudE vdNvdudMudL td vd G td vd td ud F td ud E td vd N td vd td ud M td ud L kn                                                                              2 2 2 2 2 2 2 2 22 22 22 22 - kn is independent on dt therefore on C. - kn is a function of the surface parameters L, M, N, E, F, G and of the direction . vd ud - Because I = E du2 + 2 F du dv + G dv2 > 0 → sign kn=sign II - kn is independent on coordinates since I and II are independent.  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C Differential Geometry in the 3D Euclidean Space Table of Contents
  • 70. 70 SOLO Principal Curvatures and Directions G vd ud F vd ud E N vd ud M vd ud L I II vdGvdudFudE vdNvdudMudL kn                             2 2 2 2 2 2 22 22 - kn is a function of the surface parameters L, M, N, E, F, G and of the direction .vd ud Let find the maximum and minimum of kn as functions of the directions d u/ d v.  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C If this occurs for d u0/ d v0 we must have:         0&0 00 00 0000 00 00 , 2 ,, 2 ,           vdud vdvd vdud n vdud udud vdud n I IIIIII v k I IIIIII u k               0&0 00 00 00 00 00 00 00 00 00 00 , ,, 0 , ,, 0                   vdud vdnvd vdud vdvd vdud n vdud udnud vdud udud vdud n IkII I II III v k IkIII I II II u k Multiply by I and use  00 , 0 vdud n I II k  Differential Geometry in the 3D Euclidean Space
  • 71. 71 SOLO Principal Curvatures and Directions (continue – 1)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C           0&0 00 00 00 00 00 00 , , 0 , , 0       vdud vdnvd vdud n vdud udnud vdud n IkII v k IkII u k 22 2: vdNvdudMudLNdrdII  22 2: vdGvdudFudErdrdI  00 220 vdFudEI ud  00 220 vdGudFI vd  00 220 vdMudLII ud  00 220 vdNudMII vd       0 00 00 00 , , 0    vdud udnud vdud n IkII u k      0 00 00 00 , , 0    vdud vdnvd vdud n IkII v k     00000 0  vdFudEkvdMudL n     00000 0  vdGudFkvdNudM n Differential Geometry in the 3D Euclidean Space
  • 72. 72 SOLO We found: Principal Curvatures and Directions (continue – 2)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C              0 0 0000 0000 0 0 vdGudFkvdNudM vdFudEkvdMudL n n or:                       0 0 0 0 00 00 vd ud GkNFkM FkMEkL nn nn This equation has non-trivial solution if: 0det 00 00            GkNFkM FkMEkL nn nn or expending:       02 222 00  MNLkMFLGNEkFGE nn Differential Geometry in the 3D Euclidean Space
  • 73. 73 SOLO Study of the quadratic equation: Principal Curvatures and Directions (continue – 3)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C The discriminant of this equation is:       02 222 00  MNLkMFLGNEkFGE nn     222 42 MNLFGEMFLGNE      2 22 222 2 2 2222 2 2 22222424                      E LF LG E LF MFLGNEENLLFMELF E FGE LFMELFME E FGE     NLFNLGE E MLF LMGF E LF E LGF LFME E F LGNELFME E FGE 2 3 2 24222 2 2 2 44884424               E LGF LG E LGF LMGFLG NLGE E LF E MLF E LGF NLF E LF 22 22 22 22 2 24322 2 2 24 84884 488444            024 42 2 2 2 0 2 222                      LFME E F LGNELFME E FGE MNLFGEMFLGNE  Differential Geometry in the 3D Euclidean Space
  • 74. 74 SOLO Study of the quadratic equation (continue – 1): Principal Curvatures and Directions (continue – 4)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C The discriminant of this equation is:       02 222 00  MNLkMFLGNEkFGE nn          024 42 2 2 2 0 2 222                      LFME E F LGNELFME E FGE MNLFGEMFLGNE  The discriminant is greater or equal to zero, therefore we always obtain two real solutions that give extremum for kn: 21 , nn kk Those two solutions are called Principal Curvatures and the corresponding two directions are called Principal Directions    2211 ,,, vdudvdud 0&0  LGNELFME G N F M E L  The discriminant can be zero if:     02&0  LFME E F LGNELFME In this case: G N F M E L vdGvdudFudE vdNvdudMudL kn     22 22 2 2 This point in which kn is constant in all directions is called an Umbilical Point. Differential Geometry in the 3D Euclidean Space
  • 75. 75 SOLO Gaussian and Mean Curvatures Principal Curvatures and Directions (continue – 5)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C Rewrite the equation:       02 222 00  MNLkMFLGNEkFGE nn as:         0 2 2 2 2 2 00        FGE MNL k FGE MFLGNE k nn We define:    2 2 : 21 FGE MFLGNE kkH nn       2 2 21 : FGE MNL kkK nn    Mean Curvature Gaussian Curvature Karl Friederich Gauss 1777-1855 Differential Geometry in the 3D Euclidean Space
  • 76. 76 SOLO Gaussian and Mean Curvatures (continue – 1) Principal Curvatures and Directions (continue – 6)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C    2 2 21 : FGE MNL kkK nn    Gaussian Curvature    vurvurr ,,  Change of coordinates from u,v to θ,φ    vuvv vuuu , ,  The coordinates are related by                   v u vv uu vd ud vu vu     II vd ud NM ML vdud vd ud vv uu NM ML vu vu vdudII vu vu vv uu                                        We found:     I vd ud GF FE vdud vd ud vv uu GF FE vu vu vdudI vu vu vv uu                                                                  vu vu vv uu vv uu GF FE vu vu GF FE                           vu vu vv uu vv uu NM ML vu vu NM ML   2 2 2 2 detdetdetdet                                             vu vu vu vu vv uu FGE vv uu GF FE GF FE FGE   2 2 2 2 detdetdetdet                                             vu vu vu vu vv uu MNL vv uu NM ML NM ML MNL Therefore: invariant to coordinate changes        2 2 2 2 21 : FGE MNL FGE MNL kkK nn       Differential Geometry in the 3D Euclidean Space
  • 77. 77 SOLO Principal Curvatures and Directions (continue – 7)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2CStart with:              0 0 0000 0000 0 0 vdGudFkvdNudM vdFudEkvdMudL n n rewritten as :                       0 01 00000 0000 n kvdGudFvdNudM vdFudEvdMudL that has a nontrivial solution (1,-kn0) only if: 0det 0000 0000         vdGudFvdNudM vdFudEvdMudL or:       0 2 000 2 0  vdNFMGvdudNEGLudMEFL or:       0 0 0 2 0 0                  NFMG vd ud NEGL vd ud MEFL Differential Geometry in the 3D Euclidean Space
  • 78. 78 SOLO Principal Curvatures and Directions (continue – 8)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C We obtained: This equation will define the two Principal Directions 2211 21 & vdrudrrvdrudrr vunvun                  021 21 2 2 1 1 2 2 1 1 2112212121                              vdvdG MEFL NEGL F MEFL NFMG E vdvdG vd ud vd ud F vd ud vd ud E vdvdrrvdudvdudrrududrrrr vVvuuunn        0 0 0 2 0 0                  NFMG vd ud NEGL vd ud MEFL From the equation above we have:        MEFL NFMG vd ud vd ud MEFL NEGL vd ud vd ud       2 2 1 1 2 2 1 1 Let compute the scalar product of the Principal Direction Vectors: The Principal Direction Vectors are perpendicular. Differential Geometry in the 3D Euclidean Space
  • 79. 79 SOLO Principal Curvatures and Directions (continue – 9)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C Since the two Principal Directions are orthogonal 21 21 & vdrrudrr vnun   they must satisfy the equation: Let perform a coordinate transformation to the Principal Direction:  vu,       0 2 000 2 0  vdNFMGvdudNEGLudMEFL    21 ,0&0, vdud or:   0 2 1  udMEFL   0 2 2  vdNFMG 0 NFMG 01 ud 0 MEFL 02 vd 0E 0G 0 0   NrM rrF vu vu   at P Definition: A Line of Curvature is a curve whose tangent at any point has a direction coinciding with a principal direction at that point. The lines of curvature are obtained by solving the previous differential equation Differential Geometry in the 3D Euclidean Space
  • 80. 80 SOLO Principal Curvatures and Directions (continue – 10)  vur ,  rd udru  vdrv P O N 1C k 2C k  1C 2C Suppose (du0,dv0) is a Principal Direction, then they must satisfy the equations: Rodriguez Formula NrNrL uuuu   NrNrNrM vuuvvu   NrNrN vvvv                0 0 0000 0000 0 0 vdGudFkvdNudM vdFudEkvdMudL n n              0 0 0000 0000 0 0 vdrrudrrkvdNrudNr vdrrudrrkvdNrudNr vvvunvvuv vuuunvuuu   uu rrE   vu rrF   vv rrG                  0 0 0000 0000 0 0 vvunvu uvunvu rvdrudrkvdNudN rvdrudrkvdNudN            0 0 0 0 vn un rrdkNd rrdkNd   But are in the tangent plane at P since and are, and the vectors and are independent, therefore: rdkNd n  0  Nd  rd  vr  ur  00   rdkNd n The direction (du0,dv0) is a Principal Direction on a point on a surface if and only if from some scalar k, and satisfy:00 vdNudNNd vu   00 vdrudrrd vu   rdkNd   Rodriguez Formula We found: Differential Geometry in the 3D Euclidean Space Table of Contents
  • 81. 81 SOLO Conjugate Directions  vur ,  rd udru  vdrv P O N  Q NdN   l Let P (u,v) and Q (u+du,v+dv) neighboring points on a surface. The tangent planes to the surface at p and Q intersect along a straight line L. Now let Q approach P along a given direction (du/ dv=const= PQ), then the line l will approach a limit LC. The directions PQ and LC are called Conjugate Directions. Let be the normal at P and the normal at Q.N  NdN   Let the direction of LC be given by: vrurr vu    Since LC is in both tangential planes at P and at Q we have:   0&0  NdNrNr       0 vdNudNvrurNdr vuvu           0 vdvNrvduNrudvNruduNr vvvuuvuu   We found vvuvvuuu NrNNrNrMNrL   && The previous relation becomes:   0 vdvNvduudvMuduL  Given (du,dv) there is only one conjugate direction (δu,δv) given by the previous equation. Differential Geometry in the 3D Euclidean Space Table of Contents
  • 82. 82 SOLO Asymptotic Lines The directions which are self-conjugate are called asymptotic directions. becomes:   0 vdvNvduudvMuduL  We see that the asymptotic directions are those for which the second fundamental form vanishes. Moreover, the normal curvature kn vanishes for this direction. Those curves whose tangents are asymptotic directions are called asymptotic lines. v u vd ud   If a direction (du,dv) is self-conjugate than and the equation of conjugate lines 02 22  vdNvdudMudL The conjugat and asymptotic lines were introduced by Charles Dupin in 1813 in “Dévelopments de Géométrie”. Pierre Charles François Dupin 1784 - 1873 http://www.groups.dcs.st-and.ac.uk/~history/Biographies/Dupin.html Differential Geometry in the 3D Euclidean Space Table of Contents
  • 83. 83 SOLO Vectors & Tensors in a 3D Space Scalar and Vector Fields Let express the cartesian coordinates (x, y, z) of any point, in a three dimensional space as a function of three curvilinear coordinates (u1, u2, u3), where:  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3                 zyxuuzyxuuzyxuu uuuzuuuyuuuxx ,,,,,,,, ,,,,,,,, 332211 321321321 Those functions are single valued with continuous derivatives and the correspondence between (x,y,z) and (u1,u2,u3) is unique (isomorphism). kzjyixr   or 3 3 2 2 1 1 ud u r ud u r ud u r rd               321 ,,,, uuuzyx     321 ,,,, uuuAzyxAA   Assume now that the scalars Φ and vectors are functions of local coordinates, cartesian (x,y,z) or general, curvilinear (u1,u2,u3) A  In general ( we can not assume that Φ and are functions of position). A     rAAr   , Table of Contents
  • 84. 84 SOLO Vectors & Tensors in a 3D Space Vector Differentiation  tAA  Let a vector function of a single parameter tA  Ordinary Derivative of Scalars and Vectors The Ordinary Derivative of the Vector is defined as       t tAttA td tAd t      0 lim  tA   t t A     tAttAA   If the limit exists we say that is continuous and differentiable in t. tA  Differentiation Formulas If are differentiable vector functions of a scalar t and φ is a differentiable scalar of t, then CBA  ,,   td Bd td Ad BA td d      td Bd AB td Ad BA td d        td Ad A td d A td d         td Cd BAC td Bd ACB td Ad CBA td d          td Bd AB td Ad BA td d                            td Cd BAC td Bd ACB td Ad CBA td d       Table of Contents
  • 85. 85 SOLO Vectors & Tensors in a 3D Space Vector Differentiation Partial Derivatives of Scalar and Vectors    321 ,,,, uuuzyx     321 ,,,, uuuAzyxAA   Assume now that the scalars Φ and vectors are functions of local coordinates, cartesian (x,y,z) or general, curvilinear (u1,u2,u3) A  The partial derivatives are defined as follows       1 3213211 0 1 321 ,,,, lim ,, 1 u uuuuuuu u uuu u              2 3213221 0 2 321 ,,,, lim ,, 2 u uuuuuuu u uuu u              3 3213321 0 3 321 ,,,, lim ,, 3 u uuuuuuu u uuu u              1 3213211 0 1 321 ,,,, lim ,, 1 u uuuAuuuuA u uuuA u              2 3213221 0 2 321 ,,,, lim ,, 2 u uuuAuuuuA u uuuA u              3 3213321 0 3 321 ,,,, lim ,, 3 u uuuAuuuuA u uuuA u        Higher derivatives are also defined                                                                                           2 3 2 1 2 31 3 1212 2 2121 2 33 2 3 2 22 2 2 2 11 2 1 2 && && u A uuu A u A uuu A u A uuu A u A uu A u A uu A u A uu A   Table of Contents
  • 86. 86 SOLO Vectors & Tensors in a 3D Space Vector Differentiation Differentials of Vectors 3 3 2 2 1 1 ud u A ud u A ud u A zd z A yd y A xd x A Ad                     If     321321 111111,,,, 321 uAuAuAzAyAxAuuuAzyxAA uuuzyx        321321 111111111 321321 udAudAudAuAduAduAdzAdyAdxAdAd uuuuuuzyx     BdABAdBAd     BdABAdBAd     CdBACBdACBAdCBAd           CdBACBdACBAdCBAd   then If are differentiable vector functions of a scalar t.CBA  ,, Table of Contents
  • 87. 87 SOLO Vectors & Tensors in a 3D Space The Vector Differential Operator Del (, Nabla) We define the Vector Differential Operator Del (, Nabla) in Cartesian Coordinates as: z z y y x x 111:          This operator has double properties: (a) of a vector, (b) of a differential Gradient: Nabla operates on a Scalar or Vector Field z z y y x x z z y y x x 111111:                          zz z A yz z A xz z A zy y A yy y A xy y A zx x A yx x A xx x A zAyAxAz z y y x x A zyx zyx zyx zyx 111111 111111 111111 111111:                                            a scalar a dyadic
  • 88. 88 SOLO Vectors & Tensors in a 3D Space The Vector Differential Operator Del (, Nabla) (continue) We define the Vector Differential Operator in Cartesian Coordinates as: z z y y x x 111:          This operator has double properties: (a) of a vector, (b) of a differential Divergence: Nabla performs a Scalar Product on a Vector Field   z A y A x A zAyAxAz z y y x x A zyx zyx                        111111:  Curl (Rotor): Nabla performs a Vector Product on a Vector Field   z y A x A y x A z A x z A y A AAA zyx zyx zAyAxAz z y y x x A xyzxyz zyx zyx 111 111 111111:                                                          Table of Contents
  • 89. 89 SOLO Vectors & Tensors in a 3D Space Scalar Differential Let find the differentials of:    321 ,,,, uuuzyx  3 3 2 2 1 1 ud u ud u ud u zd z yd y xd x d                     rdzzdyydxxdz z y y x x d                 111111 Since      zyxuuzyxuuzyxuu ,,,,,,,, 332211  We obtain rduudrduudrduud   332211 ,, rdu u u u u u ud u ud u ud u d                             3 3 2 2 1 1 3 3 2 2 1 1 Comparing with we obtainrdd                             3 3 2 2 1 13 3 2 2 1 1 u u u u u uu u u u u u Using the Gradient definition: z z y y x x 111:          or 3 3 2 2 1 1 : u u u u u u          in general curvilinear coordinates Table of Contents
  • 90. 90 SOLO Vectors & Tensors in a 3D Space Vector Differential Let find the differentials of:    321 ,,,, uuuAzyxAA   3 3 2 2 1 1 ud u A ud u A ud u A zd z A yd y A xd x A Ad                         ArdzAyAxA z zd y yd x xd zdz z A y z A x z A ydz y A y y A x y A xdz x A y x A x x A Ad zyx zyxzyx zyx                                                              111 111111 111   ArdA u u u u u urdrdu u A u u A u u A Ad                                      3 3 2 2 1 13 3 2 2 1 1 rduudrduudrduud   332211 ,, and 3 3 2 2 1 1 : u u u u u u            ArdAd  Therefore In Cartesian Coordinates: In General Curvilinear Coordinates using Table of Contents
  • 91. 91 Vector AnalysisSOLO     Linearity of operator          Differentiability of operator   BABA   Linearity of operator   BABA   Linearity of operator   AAA    Differentiability of operator   AAA    Differentiability of operator Differential Identities
  • 92. 92 Vector AnalysisSOLO Differential Identities              BAAB BAAB BABABA BA BA                 AAA cbacabcba   2   0 0    aa      0 0 baabaa A      
  • 93. 93 Vector AnalysisSOLO Differential Identities          ABBAABBABA        BABABA B        ABBAAB A                ABBABABABAAB BA BA                  BAABABBABA        BABABA BA        ABABBA AAA        BABABA BBB  
  • 94. 94 Vector AnalysisSOLO         Differential Identities Summary   BABA     BABA     AAA      AAA             ABBAABBABA            BAABABBABA        BAABBA       AAA  2  0    0 A     AAAAA        2/ 2 Table of Contents
  • 95. 95 SOLO Vectors & Tensors in a 3D Space Curvilinear Coordinates in a Three Dimensional Space Let express the cartesiuan coordinates (x, y, z) of any point, in a three dimensional space as a function of three curvilinear coordinates (u1, u2, u3), where:  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3                 zyxuuzyxuuzyxuu uuuzuuuyuuuxx ,,,,,,,, ,,,,,,,, 332211 321321321 Those functions are single valued with continuous derivatives and the correspondence between (x,y,z) and (u1,u2,u3) is unique (isomorphism). kzjyixr   3 3 2 2 1 1 3 333 2 222 1 111 3 3 12 2 1 1 3 3 12 2 1 1 3 3 12 2 1 1 ud u r ud u r ud u r udk u z j u y i u x udk u z j u y i u x udk u z j u y i u x kud u z d u z ud u z jud u y d u y ud u y iud u x d u x ud u x kzdjydixdrd                                                                                                             or 3 3 2 2 1 1 ud u r ud u r ud u r rd           
  • 96. 96 SOLO Vectors & Tensors in a 3D Space Curvilinear Coordinates in a Three Dimensional Space (continue – 1) 3 3 2 2 1 1 ud u r ud u r ud u r rd            Let define: 3,2,1: 1     i u r r iu   If and are linear independent (i.e. if and only if αi = 0 i=1,2,3) then they form a base of the space E3. 21 , uu rr  3u r  0 3 1  i ui i r We have also:   3,2,1,,1  irdzyxuud i  We can write:         3 1 2 1 1 11 1 321 ,, udurudurudurrdzyxuud uuu   Because du1, du2, du3 are independent increments the precedent equation requires: 001 111 321  ururur uuu  Similarly by multiplying by and we obtain:rd  2 u 3 u         ji ji uru u r j i j u j i 0 1 1    Therefore and are reciprocal systems of vectors.321 ,, uuu rrr  321 ,, uuu   dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
  • 97. 97 SOLO Vectors & Tensors in a 3D Space Curvilinear Coordinates in a Three Dimensional Space (continue – 2) We proved that reciprocal systems of vectors are related by: and are reciprocal systems of vectors.321 ,, uuu rrr  321 ,, uuu       321 21 321 13 321 32 ,, , ,, , ,, 321 uuu uu uuu uu uuu uu rrr rr u rrr rr u rrr rr u                  321 21 321 13 321 32 ,, , ,, , ,, 321 uuu uu r uuu uu r uuu uu r uuu           and    1,,,, 321 321  uuurrr uuu  or 1 ,, ,, ,, ,, 321 321 333 222 111 333 222 111                                                    zyx uuu J uuu zyx J z u y u x u z u y u x u z u y u x u u z u y u x u z u y u x u z u y u x where is the Jacobian of x,y,z with respect to u1, u2, u3.         321 ,, ,, uuu zyx J Carl Gustav Jacob Jacobi 1804 - 1851  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
  • 98. 98 SOLO Vectors & Tensors in a 3D Space Curvilinear Coordinates in a Three Dimensional Space (continue – 3)   grrr u z u y u x u z u y u x u z u y u x uuu zyx J uuu                            321 ,,det: ,, ,, 333 222 111 321  If is nonsingular the transformation from x,y,z to u1, u2, u3 is unique.        321 ,, ,, uuu zyx J  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 g rr u g rr u g rr u uuuuuu 211332 321 ,,             211332 321 ,, uugruugruugr uuu   Table of Contents
  • 99. 99 SOLO Vectors & Tensors in a 3D Space Covariant and Contravariant Components of a Vector in Base .321 ,, uuu rrr  Given a vector we have: 321 ,, uuuA      j u j j u i u i uuu urAuAuAuAuA ruArArArArAA j ii     3 3 2 2 1 1 321 321 where: juj ii rAA uAA     : : are the contravariant components of A  are the covariant components of A  The Element of Arc ds. The Metric Coefficients gij Riemann and Euler Spaces Compute:       jiuujuiu ududrrudrudrrdrdsd jiji  2 Define: jiuuuuij grrrrg ijji   the metric coefficients jijijiij ududgududgrdrdsd  2 the element of arc  dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 Leonhard Euler 1707- 1783 Georg Friedrich Bernhard Riemann 1826 - 1866A space with the metric defined above is called a Riemann Space. If gij = δij then the space is called an Euler Space.
  • 100. 100 SOLO Vectors & Tensors in a 3D Space Covariant and Contravariant Components of a Vector in Base (continue -1).321 ,, uuu rrr  or: If we substitute for we obtain:A  ju r    j ij j uu j ju ugurruAr jii                                      3 2 1 333231 232221 131211 3 2 1 u u u ggg ggg ggg r r r ugr u u u j ijui     Multiplying by gik (where gik gji = δj k) and summing on i and k: kjk j j ij ik u ik uuuggrg j    Changing k by j we obtain:                                    3 2 1 333231 232221 131211 3 2 1 u u u u ijj r r r ggg ggg ggg u u u rgu i      dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
  • 101. 101 SOLO Vectors & Tensors in a 3D Space Covariant and Contravariant Components of a Vector in Base (continue -2).321 ,, uuu rrr  Multiplying equation by we get:iu ijj rgu   i u   ij u iijji gruguu i    1 jiij uug or Now:       j ij j ij j ijui AguAgugArAA j   j iji AgA  We also found:    321321 ,,,,det 2 333231 232221 131211 uuuuuu rrrgrrr ggg ggg ggg g               dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 Table of Contents
  • 102. 102 SOLO Vectors & Tensors in a 3D Space Coordinate Transformation in Curvilinear Coordinates from those equations we obtain:                            32133 32122 32111 32133 32122 32111 ,, ,, ,, ,, ,, ,, uuuuu uuuuu uuuuu uuuuu uuuuu uuuuu Let and be two general curvilinear coordinates in an E3 space. There exists a unique transformation from one curvilinear coordinates to the other:  321 ,, uuu  321 ,, uuu k k j k k k j jj j i j j j i i ud u u ud u u udud u u ud u u ud               3 1 3 1 k k j j i j k k k j j i j j j i i ud u u u u ud u u u u ud u u ud                 3 1 3 1 3 1 Because dui and duk are independent:           ki ki u u u u i k k j j i 0 1  In the same way: k k i i j i i j j ud u u u u du u u ud         therefore:           kj kj u u u u j k k i i j 0 1   dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
  • 103. 103 SOLO Vectors & Tensors in a 3D Space Coordinate Transformation in Curvilinear Coordinates (continue – 1) If:    321321 ,,,, uuuruuurr   j i ij u u u r u r         ij u j i u r u u r     and: i j ji u u u r u r         ji u i j u r u u r     Let be a given vector with respect to two curvilinear coordinates.A  ji u j j ii u i rA u u ArAA          i j i ji u i j i u i ji u i i j j uA u u urA u u ur u u AurAuAuAA j ji                     i j ij A u u A    This is a contravariant relation with respect to the reference relation .ij u j i u r u u r     j i j i A u u A    This is a covariant relation with respect to the reference relation . ij u j i u r u u r      dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3
  • 104. 104 SOLO Vectors & Tensors in a 3D Space Coordinate Transformation in Curvilinear Coordinates (continue – 2) Let define: j i j i i A u u AuA    11:  By multiplying the last equation by and because then and: 1     i j j i u u u u j i u u   j i j u u A    j j ij j i u u u uAuA    :  j j ii u u u u     From the reference relation mjki u j m uu i k u r u u rr u u r        & mkmkji uu j m i k u j m u i k uuij rr u u u u r u u r u u rrg              kl j m i k ij g u u u u g      This is a two order covariant relation with respect to the reference relation .ij u j i u r u u r     In the same way m m jjk k ii u u u uu u u u        & mk m j k im m jk k ijiij uu u u u u u u u u u u uug             km m j k iij g u u u u g      This is a two order contravariant relation with respect to the reference relation .ij u j i u r u u r      dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 Table of Contents
  • 105. 105 SOLO Vectors & Tensors in a 3D Space Covariant Derivative First we want to find the derivatives and . j u u r i    j i u u   Because are vectors of a base in E3 we can write as a function of this base. j u u r i    3,2,1ir iu  i i uij k ijj u r u r uu r                  :        ij k ij kEquivalent notation Where are the Cristoffel’s Symbols of II kind that we must determine.ij k  Elwin Bruno Cristoffel 1829 - 1900 Because thenk ji k uji k i u jiijj u uij k r u r u r uu r uu r r                                    ji k ij k  Let calculate now  mjigrrr u r mijkmij k uuij k u j u mkm i ,: ,      where are the Cristoffel’s Symbols of I kind  kmij k u j u mij gr u r mji m i       :,, Because andji k ij k  mkkm gg  kjikij ,, 
  • 106. 106 SOLO Vectors & Tensors in a 3D Space Covariant Derivative (continue – 1) To find let perform the following calculations:kij,    ijkjki k u uu k u uu kk ij u r rr u r rr uu g j ij i ji ,,                   ijkkji j u uu j u uu jj ik u r rr u r rr uu g k ik i ki ,,                   jikkji i u uu i u uu ii jk u r rr u r rr uu g k jk j kj ,,                 From those equations we obtain:                  k ij i jk j ik kij u g u g u g 2 1 ,                  k ij i jk j ik km kij km ij m u g u g u gg g 2 , Multiplying the equations by and summing we obtain:kmij m kij g , km g The Operator .
  • 107. 107 SOLO Vectors & Tensors in a 3D Space Covariant Derivative (continue – 2) Now let find . j i u u   Because are vectors of a base in E3 we can write as a function of this base: 3,2,1 iui j i u u   Tacking the derivative with respect to uj of the equation we get: k i k u ur i   0      j k u k j u u u ru u r i i   or ij kk mij mk uij mk j u j k u uru u r u u r m i i            k jk i j i u u u    : But we have also ij k ij k iuij k j k u ur u u r ii       Therefore: i ij k j k u u u    Because we have:mu imi rgu   iuij kim j k rg u u    
  • 108. 108 SOLO Vectors & Tensors in a 3D Space Covariant Derivative (continue – 3) We found that: m jkim i kmjmikjjki k ij gg u g    ,, Let find , where . k ij u g   jiij uug  j km iki km mjkj km ijmi km k j ij k i k ij gguuuu u u uu u u u g          j km iki km mj k ij gg u g    We can see that: 0       j kj i ki j jk i ki j km ik ij i km mj ij m jkim iji kmjm ij k ijij k ij ij k j m i j m i m gggggggg u g g u g g   This can be proven if we take the derivative with respect to uk of the equation: 1ij ij gg Table of Contents
  • 109. 109 SOLO or Vectors & Tensors in a 3D Space Covariant Derivative of a Vector .A  j ju i uArAA i   iimi i i u i mk i u k i mi u m ik i u k i k ui u k i k rAr u A rAr u A u r Ar u A u A                       ji mk i k i iju i mk i k i k uA u A grA u A u A i                            But we have also or ji mk i k i ij jm jkj k j k uA u A guA u A u A                           jm jkj j k j jm mj mkj j k j k j j j k j k uAu u A uAu u A u u Au u A u A                   Therefore we can write:               i mk i k i ij m jkj k j A u A gA u A By multiplying the equation by gij and summing we obtain:               m jkj k jiji mk i k i A u A gA u A Table of Contents
  • 110. 110 Vector AnalysisSOLO Dyadic Identities Summary       CbaCabCba        CbaCabCba     CCC      CCC        CCC  2  0 C  aCCa T    TT aCCa       BCaBaC TT  
  • 111. 111 SOLO Vector Integration Vector Analysis Ordinary Integration of Vectors Let be a vector depending on the single scalar variable t, with Ax (t), Ay (t), Az (t) continuous in a specific interval, then         ztAytAxtAtA zyx 111           zdttAydttAxdttAdttA zyx 111    If there exists a vector such that then:   tS td d tA   tS        ctSdttS td d dttA    where is an arbitrary constant.c  The definite integral between t = a and t = b gives           aSbSctSdttS td d dttA b a b a b a    Table of Contents
  • 112. 112 SOLO Vector Integration                         n i iiii n i iiiiin trzyxAtrtrzyxAS 11 1 ,,,,   C 1t n tb  2 t 0 ta  1i t i t 1 2 i n  itr  Let subdivide C into n parts by n arbitrary points t1, t2,…,tn, and call a=t0 and b=tn. On each arc joining ti-1 to zi choose a point ξi. Define the sum:                 C b a n i iiii z n n rdzyxArdzyxAtrzyxAS i  ,,,,,,limlim 1 0  Properties of Integrals     CCC rdBrdArdBA  constantrdArdA CC       a b b a rdArdA    b c c a b a rdArdArdA  Vector Analysis Line Integrals Let be continuous at all points on a curve C of a finite length L, defined by the position vector . tr  zyxA ,,  Let the number of subdivisions n increase in such a way that the largest of approaches zero, then the sum approaches a limit that is called the line integral (also Riemann-Stieltjes integral).  itr   Georg Friedrich Bernhard Riemann 1826 - 1866 Table of Contents
  • 113. 113 SOLO Vector Integration Vector Analysis Surface Integrals  vur ,  rd udru  vdrv  sd P O vdrv  udru  vdrudrvdrudrsd vuvu   Let be continuous at all points on a surface S of a finite area A, defined by the position vector . vur ,   zyxA ,,  A surface integral over the vector field is defined asA             S vu S sdnsd S vdudrrvuxvuyvuxAsdnAsdA  ,,,,,ˆ ˆ Table of Contents
  • 114. 114 SOLO Vector Integration Vector Analysis Volume Integrals Let be continuous at all points on a finite volume V, defined by the position vector . 321 ,, uuur    321 ,,,, uuuAzyxA   A volume integral over the vector field is defined asA      V uuu V udududrrruuuAdvA 321321 321 ,,,,  where                                                     321 333 222 111 ,, ,, det,, 321321 uuu zyx J u z u y u x u z u y u x u z u y u x rrrrrr uuuuuu   dr constu 3 i  j  k  1 1 ud u r    2 2 ud u r    3 3 ud u r    constu 1 constu 2 curveu1 curveu2 curveu3 Table of Contents
  • 115. 115 SOLO Simply and Multiply Connected Regions A region R is called simply-connected if any simple closed curve Γ, which lies in R can be shrunk to a point without leaving R. A region R that is not simply-connected is called multiply-connected. C0 x y R C1  C0 x y R C1 C2 C3  C x y R  C x y R simply-connected multiply-connected. Vector Integration Vector Analysis Table of Contents
  • 116. 116 SOLO Green’s Theorem in the Plane C R Let P (x,y) and Q (x,y) be continuous and have continuous partial derivatives in a region R and on the boundary C. Green’s Theorem states that:                R dydx y P x Q dyQdxP C Vector Integration Vector Analysis http://en.wikipedia.org/wiki/George_Green This Theorem was first published by George Green (1793 – 1841) in 1828 in a paper “An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism”.
  • 117. 117 SOLO Green’s Theorem in the Plane Vector Integration Vector Analysis Lord Kelvin rediscovered his work four years after his death and gave it wide publicity. Kelvin, James Clerk Maxwell, George Gabriel Stokes and others built on his pioneering work and Green gained a posthumous reputation amongst 19th- and 20th-century mathematicians and scientists. His work has had great influence and nowadays he is remembered principally for Green’s theorem in vector analysis, Green’s tensor (or the Cauchy-Green tensor) in elasticity theory and above all for Green’s functions for solving differential equations. George Green (1793-1841) was one of the most remarkable of nineteenth century physicists, a self-taught mathematician whose work has contributed greatly to modern physics. He was a pioneer in the application of mathematics to physical problems. He had very little formal education and died without achieving any recognition among other mathematicians. http://www.historyoftheuniverse.com/george_green/store.htm#Vol_1_paper George Green 1793-1841 tomb stone
  • 118. 118 SOLO Proof of Green’s Theorem in the Plane C R P T S Q a b x y  xgy 2  xgy 1 Start with a region R and the boundary curve C, defined by S,Q,P,T, where QP and TS are parallel with y axis.              b a xgy Xgy dy y P dxdydx y P 2 R By the fundamental lemma of integral calculus:                  xgxPxgxPyxPdy y yxP xgy xgy xgy Xgy 12 ,,, , 2 1 2         Therefore:          b a b a dxxgxPdxxgxPdydx y P 12 ,, R but:        a bSQ dxxgxPdxxgxP 22 ,, integral along curve SQ        b aPT dxxgxPdxxgxP 11 ,, integral along curve PT If we add to those integrals:     00,,   dxsincedxyxPdxyxP QPTS we obtain:                CTSPTQPSQ dxyxPdxyxPdxxgxPdxyxPdxxgxPdydx y P ,,,,, 12 R Assume that PT is defined by the function y = g1 (x) and SQ is defined by the function y = g2 (x), both smooth and y P   is continuous in R: Vector Integration Vector Analysis
  • 119. 119 SOLO Proof of Green’s Theorem in the Plane (continue – 1)      C dxyxPdydx y P , R In the same way:      C dyyxQdydx x Q , R Therefore we obtain:                R dydx y P x Q dyQdxP C The line integral is evaluated by traveling C counterclockwise. For a general single connected region, as that described in Figure to the right, can be divided in a finite number of sub-regions Ri, each of each are of the type described in the Figure above. Since the adjacent regions boundaries are traveled in opposite directions, there sum is zero, and we obtain again:                R dydx y P x Q dyQdxP C C R4 x y R R3 R1 R2 C R P T S Q a b x y  xgy 2  xgy 1 Vector Integration Vector Analysis
  • 120. 120 SOLO Proof of Green’s Theorem in the Plane (continue – 2) The general multiply-connected regions can be transformed in a simply connected region by infinitesimal slits C0 x y R C1 P0 P1 C0 x y R C1 C2 C3                   R dydx y P x Q dyQdxPdyQdxP i CC i0 All line integrals are evaluated by traveling Ci i=0,1,… counterclockwise. Since the slits boundaries are traveled in opposite directions, there integral sum is zero:     0 0 1 1 0   P P P P dyQdxPdyQdxP We obtain: Vector Integration Vector Analysis Table of Contents
  • 121. 121 SOLO Stoke’s Theorem C R Let P (x,y) and Q (x,y) be continuous and have continuous partial derivatives in a region R and on the boundary C. Green’s Theorem states that: GEORGE STOCKES 1819-1903 A more general theorem was given by Stokes                R dydx y P x Q dyQdxP C                                      yzxzxy RRR dzdy z Q y R dzdx x R z P dydx y P x Q dzRdyQdxP C or in vector form:   S dAFdrF C  where:         zzyxRyzyxQxzyxPzyxF 1,,1,,1,,,,   zdzydyxdxdr 111  zdydxydzdxxdzdydA 111  GEORGE GREEN 1793-1841 z z y y x x 111          Vector Integration Vector Analysis
  • 122. 122 Proof of Stoke’s Theorem SOLO GEORGE STOCKES 1819-1903 Vector Analysis A B C D v u duu  dvv  Constant v curves Constant u curves  vur ,   vuA ,  C Consider a surface in a 3 dimensional space, defined by two parameters u and v and bounded by a curve Γ. Let choose four points on this surface:  vurA ,:  vduurB ,:   dvvurD ,:  dvvduurC  ,: Consider also a vector, function of the position:  vuA ,  The vector at the four points, A, B, C, D, is given by:    vuAAA ,                  vuAdu u r vuAvuArdvuAAdvuAvduuABA uu ,,,,,,                           vuAdv v r vuAvuArdvuAAdvuAdvvuADA vv ,,,,,,                       vuAdv v r vuAdu u r vuAAdAdvuAdvvduuACA vu ,,,,,                     where du, dv are infinitesimals (differentials)
  • 123. 123 Proof of Stoke’s Theorem (continue – 1) SOLO Vector Analysis    vuAAA ,        vuAdu u r vuABA ,,                 vuAdv v r vuADA ,,                   vuAdv v r vuAdu u r vuACA ,,,                     A B C D v u duu  dvv  Constant v curves Constant u curves  vur ,   vuA ,  C Let compute the path integral:                 dvdu u r Adv v r v r A u r dv v r Adv v r Adu u r Adv v r Adu u r A dv v r Adv v r Adu u r Adu u r Adu u r A dr AADA dr DACA dr CABA dr BAAA drA DACDBCAB ABCD                                                                                                                                                                       2 1 2 1 2 1 2 1 2222
  • 124. 124 Proof of Stoke’s Theorem (continue – 2) SOLO Vector Analysis A B C D v u duu  dvv  Constant v curves Constant u curves C  vur ,   vuA ,  Let compute: dvdu u r A v r v r A u r drA ABCD                                                        u r A v r v r A u r v r u r AA u r v r u r A v r u r A A                                                                                                 Therefore:     sdAdvdu v r u r AdrA ABCD              We identify as the vector describing the surface ABCD since it is normal to surface and the aria is equal to dvdu v r u r sd               : dvdu v r u r             Let sum over the entire u,v network. Interior line integrals will cancel out in pairs leaving only , and finally we obtain:   C drA     SC sdAdrA  Table of Contents
  • 125. 125 Divergence Theorem This therem is also known as Gauss’ Theorem, Ostrogradsky’s Theorem or Gauss-Ostrogradsky Theorem. V A   ds SOLO Vector Analysis JOSEPH-LOUIS LAGRANGE 1736-1813 The theorem was first discovered by Lagrange in 1762, than later rediscovered by Carl Friedrich Gauss in 1813,by George Green in 1825, and in 1831 by Michail Vasilievich Ostrogradsky who gave the first proof. MIKHAIL VASILIEVICH OSTROGRADSKI 1801-1862 GEORGE GREEN 1793-1841 tomb stone http://en.wikipedia.org/Divergence_theorem JOHANN CARL FRIEDRICH GAUSS 1777-1855    S V dvAsdA 
  • 126. 126 Proof of Divergence Theorem SOLO Vector Analysis    S V dvAsdA         2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u     2 , 2 , 2 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu dV is any volume, that includes the point (u1,u2,u3) and is closed by the surface S=S1+S2+S3+S4+S5+S6   654321 SSSSSSS AdsAdsAdsAdsAdsAdsAds  where 32 32 3 3 2 2 41 udud u r u r ud u r ud u r dsds                      31 13 1 1 3 3 52 udud u r u r ud u r ud u r dsds                      21 21 2 2 1 1 63 udud u r u r ud u r ud u r dsds                        321 321 3 3 2 2 1 1 321 1 1 321 2 ,, 2 ,, 41 ududud u r u r u A ud u r ud u rud u A uuuA ud u A uuuAAdsAds SS                                                                      321 132 1 1 3 3 2 2 321 2 2 321 2 ,, 2 ,, 52 ududud u r u r u A ud u r ud u rud u A uuuA ud u A uuuAAdsAds SS                                                                      321 213 2 2 1 1 3 3 321 3 3 321 2 ,, 2 ,, 63 ududud u r u r u A ud u r ud u rud u A uuuA ud u A uuuAAdsAds SS                                                             
  • 127. 127 SOLO Vectors & Tensors in a 3D Space        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu   654321 SSSSSSS AdsAdsAdsAdsAdsAdsAds  321 213132321 ududud u r u r u A u r u r u A u r u r u A                                                          321 321 ududud u r u r u r vd                   Proof of Divergence Theorem (continue – 1)      321 21 321 13 321 32 ,, , ,, , ,, 321 uuu uu uuu uu uuu uu rrr rr u rrr rr u rrr rr u             Using the relations:                   VS vd u A u u A u u A uAds 3 3 2 2 1 1   We obtain: We also obtained the definition of Nabla ( ): 3 3 2 2 1 1 : u u u u u u          from which:   VS vdAAds  q.e.d. Table of Contents
  • 128. 128 VECTOR NOTATION CARTESIAN TENSOR NOTATION Gauss’ Theorem Variations   A analytic inV       A C C const vector .     S V dvsdGAUSS   2  analytic inV   S V k k dv s ds         S V dvAsdAGAUSS  1   S V k k kk dv x A dsA   SOLO Vector Analysis V A   ds Karl Friederich Gauss 1777-1855
  • 129. 129 VECTOR NOTATION CARTESIAN TENSOR NOTATION         S V dvAsdAGAUSS  3    V dvAA    ,A  analytic inV     S V k k kk dv x A dsA              V k k k k dv x A x A              B e e e  1 1 2 2 3 3          S V dvABBAsdABGAUSS  4            S V k k i k i kkki dv x A B x B AdsAB        A analytic inV     S V dvAAsdGAUSS  5             S V j i i j ijji dv x A x A AdsAds     SOLO Vector Analysis Table of Contents Gauss’ Theorem Variations (continue)
  • 130. 130 CS A   ds  dr VECTOR NOTATION CARTESIAN TENSOR NOTATION Stokes’ Theorem Variations   SC sdArdAStokes  1    A analytic on S           S k j i i j C ii sd x A x A rdA     GEORGE STOCKES 1819-1903 SOLO Vector Analysis      SSC sdAAsdArdAStokes  2   SC sdrdStokes   3     AsdsdAA const    
  • 131. 131 SOLO Vector Analysis vectorconstCCAA .       SC sdCArdCA             CsdAnCsdAnnAC sdACsdACArdC SS sdnsd SSC         ˆˆˆ ˆ     A d r A d s C S          SC sdAnArdStokes  ˆ4      ACACCA constC         ArdCrdCA              AnnAAn AnnAAn A A       ˆˆˆ ˆˆˆ        AnAnAnAn   ˆˆˆˆ           SSC sdAnAnAnsdAnArdStokes  ˆˆˆˆ4 Stokes’ Theorem Variations (continue - 1)
  • 132. 132 GAUSS’AND STOKES’ THEOREMS ARE GENERALIZATIONS OF THE FUNDAMENTAL THEOREM OF CALCULUS  A b A a d A x d x d x a b ( ) ( )   SOLO Vector Analysis    SC sdAnArdStokes  ˆ4 Use with rA         nnnrnrnrn r ˆ2ˆ3ˆˆˆˆ 3     therefore   SC sdnrdrStokes ˆ 2 1 5  Stokes’ Theorem Variations (continue - 2) Table of Contents
  • 133. 133 SOLO GREEN’s IDENTITIES Start from Gauss’ Theorem that relates the integral of the flux of a union of closed surfaces to it’s divergence. n i iSS 1          S GAUSS V dSnFGdvFG 1   n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   and must be continuous and twice differentiable in V.G  F  Using the identity we obtain         FGFGFG                S GAUSS V dSnFGdvFGFG 1  First Vector Green Identity Interchanging and we obtainG  F               S GAUSS V dSnGFdvGFFG 1  By subtracting the second identity from the first we obtain               SV dSnGFFGdvFGGF 1  Second Vector Green Identity Vector Analysis GEORGE GREEN 1793-1841 Table of Contents Harmonic
  • 134. 134 SOLO Derivation of Nabla (  ) from Gauss’ Theorem Start from Gauss’ Theorem that relates the integral of the flux of a union of closed surfaces to it’s divergence. n i iSS 1   n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Vector Analysis                             ,, 1 lim: 1 lim, 1 lim, 1 lim, 0 0 0 0 S V S V F S VF S V F ds V Ads V trA Ads V trA ds V tr                             S V F S VF S V F Ads V trA Ads V trA ds V tr    1 lim, 1 lim, 1 lim, 0 0 0                 S GAUSS V S GAUSS V S GAUSS V AdsvdA AdsvdA dsvd   5 1 2 Table of Contents
  • 135. 135 SOLO Vectors & Tensors in a 3D Space The Operator .                             ,, 1 lim: 1 lim, 1 lim, 1 lim, 0 0 0 0 S V S V F S VF S V F ds V Ads V trA Ads V trA ds V tr   We know that where V is any volume, that includes the point and is closed by the surface SFr         2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu Let apply those definitions to the infinitesimal volume in the figure, having the point at it’s centerFr  where 321 321 3 3 2 2 1 1 ududud u r u r u r ud u r ud u r ud u r V                                        S VF ds V tr 1 lim, 0  Gradient V is any volume, that includes the point (u1,u2,u3) and is closed by the surface S=S1+S2+S3+S4+S5+S6
  • 136. 136 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 1) where        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu   654321 SSSSSSS dsdsdsdsdsdsds 32 32 3 3 2 2 41 udud u r u r ud u r ud u r dsds                      31 13 1 1 3 3 52 udud u r u r ud u r ud u r dsds                      21 21 2 2 1 1 63 udud u r u r ud u r ud u r dsds                        321 321 3 3 2 2 1 1 321 1 1 321 2 ,, 2 ,, 41 ududud u r u r u ud u r ud u rud u uuu ud u uuudsds SS                                                              321 132 1 1 3 3 2 2 321 2 2 321 2 ,, 2 ,, 52 ududud u r u r u ud u r ud u rud u uuu ud u uuudsds SS                                                              321 213 2 2 1 1 3 3 321 3 3 321 2 ,, 2 ,, 63 ududud u r u r u ud u r ud u rud u uuu ud u uuudsds SS                                                     
  • 137. 137 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 2) or        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu   654321 SSSSSSS dsdsdsdsdsdsds 321 213132321 ududud u r u r uu r u r uu r u r u                                                      321 321 ududud u r u r u r V                                                                            321 321213132 0 1 lim u r u r u r uu r u r uu r u r uu r u r ds V S V   3 3 2 2 1 1 u u u u u u          To the same result we could arrive using: rdu u u u u u rdu u rdu u rdu u ud u ud u ud u rdd                                      3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 Gradient
  • 138. 138 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 3) or        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u        2 , 2 , 2 3 3 2 2 1 1 ud u ud u ud u  321 ,, uuu 1ur  2ur  3u r  S4 S1 S2 S5 S3 S6 F r   321 ,, uuu                                                          321 321213132 0 1 lim u r u r u r u A u r u r u A u r u r u A u r u r Ads V A S V    3 3 2 2 1 1 u A u u A u u A uA            By the same procedure as before: Divergence    S V Ads V A  1 lim 0 or                                                          321 321213132 0 1 lim u r u r u r u A u r u r u A u r u r u A u r u r Ads V A S V    3 3 2 2 1 1 u A u u A u u A uA            By the same procedure as before: Rotor    S V Ads V A  1 lim 0 Summarize 3 3 2 2 1 1 u u u u u u         
  • 139. 139 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 4) Let develop those equations in curvilinear coordinates: Therefore:   i i u       j ij j iji u gg    We obtain:   ii u i ij u i ij i i r u ggr u gu u uuu 1,, 321           where: g r r r r i i i i u u u u    1 Second Proof:           j j jij i j ji i ud u udgud u r u r u r u r ud u r ud u r ud u r rdd                                        3 32 1 1 3 3 2 2 1 1   j iji u g     ij i j g u      i i u u u u u u u uuuu             3 3 2 2 1 1321 ,,Gradient
  • 140. 140 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 5) Let develop those equations in curvilinear coordinates: j ju i uArAA i   Divergence     i ij ju i u A uuArAA i      But j m m ij i j u mj mi i j i uA u A rA u A u A j                            Therefore   ij ij j g ji m m ji i j u imj im i j uuA u A ruA u A A                         or ij m m ji i jmi im i i gA u A A u A A                        
  • 141. 141 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 6) Divergence (continue – 1)j ju i uArAA i   ij m m ij i jmi im i i gA u A A u A A                         Let compute i im konsummationggGggG ki iki jkj ik   ik ik G g g    m kiki m kiik m ki kim u g gg u g G u g g g u g                                                  k im i mk m ik ik mi i im mj k ij i jk j ik km kij km ij m u g u g u gg u g u g u gg g 22 ,Start from But k miki gg k imki ki i mkik u g g u g g u g g miim          m ik ki mi i u gg    2 We found mmm ik ki mi i u g gu g gu gg          1 2 1 2   i i i ii iim m mi i mi im i i u Ag g A u g gu A A u g gu A A u A A                    111   ij m m ij i jmi im i i i i gA u A A u A u Ag g A                           1
  • 142. 142 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 5) Rotor     i ij ju i u A uuArAA i      But j m m ij i j u mj mi i j i uA u A rA u A u A j                            Therefore ji m m ji i j u imj im i j uuA u A ruA u A A j                         j ju i uArAA i   Let develop those equations in curvilinear coordinates:       otherwise ofnpermutatiocyclicakji ofnpermutatiocyclicakji r g uu kjiu kjiji k 0 3,1,2,,1 3,2,1,,1 ,, ,,    1,2,3ofnspermutatiocyclicarekj,i, g r u A u A g r A u A A u A A kji u j i i j kji u m m ij j i m m ji i j k k                                                ,, ,,    Use the fact that and are reciprocal vectors we have i uju r 
  • 143. 143 SOLO Vectors & Tensors in a 3D Space The Operator (continue – 6) Laplacian Δ j ju i uArAA i   Let develop those equations in curvilinear coordinates:    ij m m ij ji j ji ii i g uuu u gg ugu g g                               2 2 11   i i u       j ij j iji u gg    Using   ij m m ij i jmi im i i i i gA u A A u A u Ag g A                           1   ii u i ij u i ij i i r u ggr u gu u uuu 1,, 321           We found Table of Contents
  • 144. 144 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space Let express the cartesiuan coordinates (x, y, z) of any point, in a three dimensional space as a function of three curvilinear coordinates (u1, u2, u3), where:  dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3                 zyxuuzyxuuzyxuu uuuzuuuyuuuxx ,,,,,,,, ,,,,,,,, 332211 321321321 Those functions are single valued with continuous derivatives and the correspondence between (x,y,z) and (u1,u2,u3) is unique (isomorphism). kzjyixr   kzdjydixdrd   For orthogonal coordinates we have: 3332221113 3 2 2 1 1 eudheudheudhud u r ud u r ud u r rd                    ji ji ee ji 0 1      2 3 2 3 2 2 2 2 2 1 2 1 2 udhudhudhrdrdsd   3213213 3 2 2 1 1 udududhhhud u r ud u r ud u r Vd                   33 3 22 2 11 1 /:/:/: u r u r e u r u r e u r u r e                     
  • 145. 145 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space (continue – 1)  dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3 General Coordinates:       3 3 321 22113 2 2 321 11332 1 1 321 33221 321 21 321 13 321 32 ,, , ,, , ,, h e hhh eheh rrr rr u h e hhh eheh rrr rr u h e hhh eheh rrr rr u uuu uu uuu uu uuu uu                     33 3 22 2 11 1 321 ,, eh u r reh u r reh u r r uuu                  i i u u     Orthogonal Coordinates: i ii e uh     1 Gradient: j ju i uri  AAA    321321 ,, hhhrrrg uuu   33 3 22 2 11 1 uh e uh e uh e          
  • 146. 146 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space (continue – 2)  dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3 General Coordinates:       3 3 321 22113 2 2 321 11332 1 1 321 33221 321 21 321 13 321 32 ,, , ,, , ,, h e hhh eheh rrr rr u h e hhh eheh rrr rr u h e hhh eheh rrr rr u uuu uu uuu uu uuu uu                     33 3 22 2 11 1 321 ,, eh u r reh u r reh u r r uuu                  Orthogonal Coordinates:    321 33221133 3 3 22 2 2 11 1 1 3 33 2 22 1 11 3 3 33 2 2 22 1 1 11332211 /// uuu rhArhArhAeh h A eh h A eh h A uhAuhAuhA h e hA h e hA h e hAeAeAeA          321 321 AAA AAA A     i i u Ag g    1 ADivergence:                      3 3 21 2 2 31 1 1 32 321 1 u Ahh u Ahh u Ahh hhh A j ju i uri  AAA    321321 ,, hhhrrrg uuu  
  • 147. 147 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space (continue – 3) General Coordinates:       3 3 321 22113 2 2 321 11332 1 1 321 33221 321 21 321 13 321 32 ,, , ,, , ,, h e hhh eheh rrr rr u h e hhh eheh rrr rr u h e hhh eheh rrr rr u uuu uu uuu uu uuu uu                     33 3 22 2 11 1 321 ,, eh u r reh u r reh u r r uuu                  Orthogonal Coordinates:Rotor: 1,2,3ofnspermutatiocyclicarekj,i, g r u A u A kji u j i i j k                ,,  A j ju i uri  AAA     321 33221133 3 3 22 2 2 11 1 1 3 33 2 22 1 11 3 3 33 2 2 22 1 1 11332211 /// uuu rhArhArhAeh h A eh h A eh h A uhAuhAuhA h e hA h e hA h e hAeAeAeA          321 321 AAA AAA A     321321 ,, hhhrrrg uuu   332211 321 332211 AhAhAh uuu eheheh         A  dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3
  • 148. 148 SOLO Vectors & Tensors in a 3D Space Orthogonal Curvilinear Coordinates in a Three Dimensional Space (continue – 4) General Coordinates:       3 3 321 22113 2 2 321 11332 1 1 321 33221 321 21 321 13 321 32 ,, , ,, , ,, h e hhh eheh rrr rr u h e hhh eheh rrr rr u h e hhh eheh rrr rr u uuu uu uuu uu uuu uu                     33 3 22 2 11 1 321 ,, eh u r reh u r reh u r r uuu                  Orthogonal Coordinates: Laplacian: j ju i uri  AAA    321321 ,, hhhrrrg uuu    dr constu 3 i  j  k  1 1 111 ud u r eudh      2 2 222 ud u r eudh      3 3 333 ud u r eudh      constu 1 constu 2 curveu1 curveu2 curveu3              j ji i u gg ug 1       ji jih h e h e uug i j j i ijiji 0 /1 2                                          33 21 322 31 211 32 1321 1 uh hh uuh hh uuh hh uhhh Table of Contents
  • 149. 149 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems       z y x z y x          1. Gradient  • Cartesian:       zyx zyx 111         z r r z r            1 • Cylindrical:       zr zr 111                       sin 1 1 r r r r • Spherical:         111   rr
  • 150. 150 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 2. Divergence  • Cartesian: z A y A x A A zyx           • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      z AA r Ar rr A z r           11   111   AAAA rr                    A r A r Ar rr A r sin 1 sin sin 11 2 2 
  • 151. 151 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 3. Laplacian 2  • Cartesian: 2 2 2 2 2 2 2 zyx          • Cylindrical: • Spherical: 2 2 2 2 22 2 2 2 2 2 2 2 1111 zrrrrzrr r rr                                      2 2 222 2 2 2 sin 1 sin sin 11                        rrr r rr
  • 152. 152 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 4. Curl  • Cartesian:       y A x A A x A z A A z A y A A xy z zx y yz x                      • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      111   AAAA rr        zyx zyx AAAA 111                                         r z zr z r A Ar rr A r A z A A z AA r A 1 1          zr zr AAAA 111                                                        r r r A Ar rr A Ar r A r A A A r A 1 sin 1 sin sin 1            111   AAAA rr 
  • 153. 153 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 5. Scalar Product • Cartesian: • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      111   AAAA rr  zyx zyx BBBB 111    zzyyxx BABABABA   zr zr BBBB 111       111   BBBB rr  zzrr BABABABA     BABABABA rr  
  • 154. 154 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 6. Vector Product • Cartesian: • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      111   AAAA rr  zyx zyx BBBB 111          zyx zyx BABABABA 111    zr zr AAAA 111       111   AAAA rr        xyyxz zxxzy yzzyx BABABA BABABA BABABA             zr zr BABABABA 111           rrz zrrz zzr BABABA BABABA BABABA                rr rr r BABABA BABABA BABABA                  111   BABABABA rr 
  • 155. 155 Vector AnalysisSOLO Vector Operations in Various Coordinate Systems 7. Material Derivative • Cartesian: • Cylindrical: zr zr AAAA 111     • Spherical: zyx zyx AAAA 111      111   AAAA rr  zyx zyx vvvv 111    z A v y A v x A v t A tD AD z A v y A v x A v t A tD AD z A v y A v x A v t A tD AD z z z y z x z z y z y y y x y y x z x y x x x x                                                               Av t A tD AD       zr zr vvvv 111       111   vvvv rr  z A v A r v r A v t A tD AD z A v A r v r A v t A tD AD z A v A r v r A v t A tD AD z z zz r z z zr r z rr r r r                                                                                                                                     A r vA r v r A v t A tD AD A r vA r v r A v t A tD AD A r vA r v r A v t A tD AD r r rrr r r r sin sin sin    Table of Contents
  • 156. 156 Vector AnalysisSOLO Applications Fundamental Theorem of Vector Analysis for a Bounded Region V (Helmholtz’s Theorem) Reynolds’ Transport Theorem Poisson’s Non-homogeneous Differential Equation Kirchhoff’s Solution of the Scalar Helmholtz Non-homogeneous Differential Equation Table of Contents Fundamental Theorem of Vector Analysis for a Unbounded Region V (Helmholtz’s Theorem) Laplace Fields Harmonic Functions Rotations
  • 157. 157 ROTATIONS Rotation of a Rigid Body SOLO 23r 31r 12r1 3 2 P P 1 2 331r 23r 12r A rigid body in mechanics is defined as a system of mass points subject to the constraint that the distance between all pair of points remains constant through the motion. To define a point P in a rigid body it is enough to specify the distance of this point to three non-collinear points. This means that a rigid body is completely defined by three of its non-collinear points. Since each point, in a three dimensional space is defined by three coordinates, those three points are defined by 9 coordinates. But the three points are constrained by the three distances between them: 313123231212 && constrconstrconstr  Therefore a rigid body is completely defined by 9 – 3 = 6 degrees of freedom. This is a part of the Presentation “ROTATIONS” NOTES ON ROTATIONS SOLO HERMELIN INITIAL INTERMEDIATE FINAL
  • 158. 158 ROTATIONS Rotation of a Rigid Body (continue – 1) SOLO We have the following theorems about a rigid body: Euler’s Theorem (1775) The most general displacement of a rigid body with one point fixed is equivalent to a single rotation about some axis through that point. Chasles’ Theorem (1839) The most general displacement of a rigid body is a translation plus a rotation. Leonhard Euler 1707-1783 Michel Chasles 1793-1880
  • 159. 159 ROTATIONS Rotation of a Rigid Body (continue – 2) SOLO Proof of Euler’s Theorem P 'P OA 'A B 'BC C r rr r r   O – Fixed point in the rigid body A,B – Two point in the rigid body at equal distance r from O.        rOBOA __________ A’,B’ – The new position of A,B respectively. Since the body is rigid rOBOA  __________ '' Therefore A,B, A’,B’ are one a sphere with center O.  – plane passing through O such that A and A’ are at the same distance from it.  – plane passing through O such that B and B’ are at the same distance from it. PP’ – Intersection of the planes and  The two spherical triangles APB and A’PB’ are equal. The arcs AA’ and BB’ are equal. That means that rotation around PP’ that moves A to A’ will move B to B’. q.e.d.
  • 160. 160 ROTATIONS Mathematical Computation of a Rotation SOLO A B C O   nˆ v  1v  We saw that every rotation is defined by three parameters: • Direction of the rotation axis , defined by by two parameters.nˆ • The angle of rotation , defines the third parameter. Let rotate the vector around by a large angle , to obtain the new vector   OAv  nˆ   OBv1  From the drawing we have:   CBACOAOBv1  vOA      cos1ˆˆ   vnnAC  Since direction of is:      sinˆˆ&ˆˆ  vnnvnn  and it’s length is: AC    cos1sin v   sinˆ vnCB    Since has the direction and the absolute value CB  vn  ˆ sinsinv       sinˆcos1ˆˆ1 vnvnnvv  
  • 161. 161 ROTATIONS Computation of the Rotation Matrix SOLO We have two frames of coordinates A and B defined by the orthogonal unit vectors and AAA zyx ˆ,ˆ,ˆ  BBB zyx ˆ,ˆ,ˆ The frame B can be reached by rotating the A frame around some direction by an angle .nˆ  We want to find the Rotation Matrix that describes this rotation from A to B.  ,ˆ33 nRC x B A                    sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ AAAB AAAB AAAB znznnxz ynynnxy xnxnnxx    Let write those equations in matrix form.                                               0 0 1 sinˆ 0 0 1 cos1ˆˆ 0 0 1 ˆ  AAAA B nnnx                  0 0 0 ˆ xy xz yz A nn nn nn n   0ˆ ntrace Axˆ Azˆ Ayˆ Bzˆ Byˆ Bxˆ O nˆ     Rotation Matrix
  • 162. 162 ROTATIONS Computation of the Rotation Matrix (continue – 1) SOLO Axˆ Azˆ Ayˆ Bzˆ Byˆ Bxˆ O nˆ                                                   0 0 1 sinˆ 0 0 1 cos1ˆˆ 0 0 1 ˆ  AAAA B nnnx                                               0 1 0 sinˆ 0 1 0 cos1ˆˆ 0 1 0 ˆ  AAAA B nnny                                               1 0 0 sinˆ 1 0 0 cos1ˆˆ 1 0 0 ˆ  AAAA B nnnz                   A A A B AAA x A B xCnnnIx ˆ 0 0 1 sinˆcos1ˆˆˆ 33                                A A A B AAA x A B yCnnnIy ˆ 0 1 0 sinˆcos1ˆˆˆ 33                                A A A B AAA x A B zCnnnIz ˆ 1 0 0 sinˆcos1ˆˆˆ 33              Rotation Matrix (continue – 1)
  • 163. 163 ROTATIONS Computation of the Rotation Matrix (continue – 2) SOLO Ax Az Ay Bz By Bx  O nˆ                      ,ˆsinˆcos1ˆˆ 3333 nRnnnICC x AAA x A B A B  The matrix has the following properties:  A nˆ        ATA nn  ˆˆ                                                   22 22 22 0 0 0 0 0 0 ˆˆ yxzyzx zyzxyx zxyxyz xy xz yz xy xz yz AA nnnnnn nnnnnn nnnnnn nn nn nn nn nn nn nn T x zzyzx zyyyx zxyxx nnI nnnnn nnnnn nnnnn ˆˆ 000 010 001 33 2 2 2                                  213ˆˆ  AA nntrace        nn nn nn nn nnnnn xy xz yz zyx AT ˆˆ000 0 0 0 ˆˆ                                     AATAAT x AAA nnnnnnnnInnn  ˆˆˆˆˆˆˆˆˆˆˆ 22                    T x AAAAAA nnInnnnnn ˆˆˆˆˆˆˆˆ 33  skew-symmetric Rotation Matrix (continue – 2)
  • 164. 164 ROTATIONS Computation of the Rotation Matrix (continue – 3) SOLO Ax Az Ay Bz By Bx  O nˆ                                              B Axx AAA x TATATA x TA B CnRnR nnnI nnnIC       ,ˆ,ˆ sinˆcos1ˆˆ sinˆcos1ˆˆ 3333 33 33 Note The last term can be writen in matrix form as Therefore In the same way End Note In fact is the matrix representation of the vector product:  vnn   ˆˆ    vInnvvnn x T  33 ˆˆˆˆ          vvnnnnvvnnvnn   ˆˆˆˆˆˆˆˆ    T x nnInn ˆˆˆˆ 33              nnnnvnvvnnnvnnn ˆˆˆˆˆˆˆˆˆˆˆ                nnnnnnvnnvnnnn ˆˆˆˆˆˆˆˆˆˆˆˆ  Rotation Matrix (continue – 3)
  • 165. 165 ROTATIONS Computation of the Rotation Matrix (continue – 4) SOLO Ax Az Ay Bz By Bx  O nˆ                                            sin0cos123 sinˆcos1ˆˆ sinˆcos1ˆˆ 33 33    AAA x TATATA x B A ntracenntraceItrace nnnItracetraceC Therefore cos21 B ACtrace Let compute the trace (sum of the diagonal components of a matrix) of B AC Also we have                                              sinˆcos1ˆˆcos sinˆcos1ˆˆ sinˆcos1ˆˆ 33 3333 33 AT x AT xx TATATA x B A nnnI nnnII nnnIC    sin 0 0 0 cos1cos 000 010 001 2 2 2                                       xy xz yz zzyzx zyyyx zxyxx nn nn nn nnnnn nnnnn nnnnn Rotation Matrix (continue – 4)
  • 166. 166 ROTATIONS Computation of the Rotation Matrix (continue – 5) SOLO Ax Az Ay Bz By Bx  O nˆ     Therefore we have                                      cos1cossincos1sincos1 sincos1cos1cossincos1 sincos1sincos1cos1cos 2 2 2 zxzyyzx xzyyzyx yzxzyxx B A nnnnnnn nnnnnnn nnnnnnn C We get  1 2 1 cos  B AtraceC two solutions for  If ; i.e. we obtain0sin   ,0       sin2/2,33,2 B A B Ax CCn        sin2/3,11,3 B A B Ay CCn        sin2/1,22,1 B A B Az CCn  Rotation Matrix (continue – 5)
  • 167. 167 ROTATIONS Consecutive Rotations SOLO - Perform first a rotation of the vector , according to the Rotation Matrix to the vector . v   1133 ,ˆ nR x 1v  - Perform a second a rotation of the vector , according to the Rotation Matrix to the vector . 1v   2233 ,ˆ nR x 2v   vnRv x  11331 ,ˆ         vnRvnRnRvnRv xxxx   ,ˆ,ˆ,ˆ,ˆ 3311332233122332  The result of those two consecutive rotation is a rotation defined as:      1133223333 ,ˆ,ˆ,ˆ  nRnRnR xxx  Let interchange the order of rotations, first according to the Rotation Matrix and after that according to the Rotation Matrix .  2233 ,ˆ nR x  1133 ,ˆ nR x The result of those two consecutive rotation is a rotation defined as:    22331133 ,ˆ,ˆ  nRnR xx Since in general, the matrix product is not commutative        2233113311332233 ,ˆ,ˆ,ˆ,ˆ  nRnRnRnR xxxx  Therefore, in general, the consecutive rotations are not commutative. Rotation Matrix (continue – 6)
  • 168. 168 ROTATIONSSOLO INITIALINITIAL INTERMEDIATEINITIAL INTERMEDIATE FINAL Consecutive Rotations of a DiceRotation Matrix (continue – 7)
  • 169. 169 ROTATIONS Decomposition of a Vector in Two Different Frames of Coordinates SOLO We have two frames of coordinate systems A and B, with the same origin O. We can reach B from A by performing a rotation. Let describe the vector in both frames.v  Axˆ Azˆ Ayˆ Bxˆ Bzˆ Byˆ v  O xAv zAv yAv xBv zBv yBv BzBByBBxBAzAAyAAxA zvyvxvzvyvxvv  111111               zA yA xA A v v v v               zB yB xB B v v v v  &                   BBABBABBAA BBABBABBAA BBABBABBAA zzzyyzxxzz zzyyyyxxyy zzxyyxxxxx ˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ1ˆˆ                          zABBABBABBA yABBABBABBA xABBABBABBA vzzzyyzxxz vzzyyyyxxy vzzxyyxxxxv ˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ1ˆ     from which Rotation Matrix (continue – 8)
  • 170. 170 ROTATIONS Decomposition of a Vector in Two Different Frames of Coordinates (continue – 1) SOLO                                                     zA yA xA BABABA BABABA BABABA zB yB xB v v v zzzyzx yzyyyx xzxyxx v v v ˆˆˆˆˆˆ ˆˆˆˆˆˆ ˆˆˆˆˆˆ Axˆ Azˆ Ayˆ Bxˆ Bzˆ Byˆ v  O xAv zAv yAv xBv zBv yBv   AB A B vCv   where is the Transformation Matrix (or Direction Cosine Matrix – DCM) from frame A to frame B. B AC                                BABABA BABABA BABABA B A B A zzzyzx yzyyyx xzxyxx CC ˆˆˆˆˆˆ ˆˆˆˆˆˆ ˆˆˆˆˆˆ : In the same way        BA B BB A A vCvCv   1 therefore  1  B A A B CC Rotation Matrix (continue – 9)
  • 171. 171 ROTATIONS Decomposition of a Vector in Two Different Frames of Coordinates (continue – 2) SOLO Axˆ Azˆ Ayˆ Bxˆ Bzˆ Byˆ v  O xAv zAv yAv xBv zBv yBv                    ATAAB A TB A TAAB A TAB A BTB vvvCCvvCvCvvv  2 Since the scalar product is independent of the frame of coordinates, we have      1  B A TB A B A TB A CCICC                                                                           100 010 001 3,33,23,1 2,32,22,1 1,31,21,1 3,32,31,3 3,22,21,2 3,12,11,1 B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A TB A CCC CCC CCC CCC CCC CCC CC or            3,2,10 3,2,11 ,, 3 1 jji iji kjCkiC ij k B A B A  Those are 9 equations in , but by interchanging i with j we get the same conditions, therefore we have only 6 independent equations.   3,2,1,, jijiC B A We see that the Rotation Matrix is ortho-normal (having real coefficients and the rows/columns are orthogonal to each other and of unit absolute value. Rotation Matrix (continue – 10) This means that the relation between the two coordinate systems is defined by 9 – 6 = 3 independent parameters.
  • 172. 172 ROTATIONS Differential Equations of the Rotation Matrices SOLO We want to develop the differential equation of the Rotation Matrix as a function of the Angular Velocity of the Rotation. Let define by: -the Rotation Matrix that defines a frame of coordinates B at the time t relative to some frame A.  tC B A -the Rotation Matrix that defines the frame of coordinates B at the time t+Δt relative to some frame A.  ttC B A    ,ˆ33xR -the Rotation Matrix from the frame of coordinates B at the time t to B at time t+Δt relative to some frame A.      tCRttC B Ax B A   ,ˆ33 and                                                   2 cos 2 sinˆ2 2 sinˆˆ2 sinˆcos1ˆˆ,ˆ 2 33 3333      x xx I IR Rotation Matrix (continue – 11)
  • 173. 173 ROTATIONS Differential Equations of the Rotation Matrices (continue – 1) SOLO Let differentiate the Rotation Matrix                      tC dt dIR tC t IR tC t IR t tCtCR t tCttC t C dt dC B A xxB A xx t B A xx t B A B Ax t B A B A t B A t B A                                                3333 0 3333 0 3333 0 33 0 00 ,ˆ lim ,ˆ lim ,ˆ lim ,ˆ lim limlim                                                                                  ˆ 2 cos 2 2 sin ˆ 2 2 2 sin ˆˆlim ,ˆ lim 2 2 0 3333 0 xx IR and Therefore      tC dt d dt tdC B A B A    ˆ Rotation Matrix (continue – 12)
  • 174. 174 ROTATIONS Differential Equations of the Rotation Matrices (continue – 2) SOLO The final result of the Rotation Matrix differentiation is: Since defines the unit vector of rotation and the rotation rate from B at time t to B at time t+Δt, relative to A, then is the angular velocity vector of the frame B relative to A, at the time t ˆ dt d   ˆ dt d      ˆ dt dB AB          tCt dt tdC B A B AB B A    By changing indixes A and B we obtain        tCt dt tdC A B A BA A B    Rotation Matrix (continue – 13)
  • 175. 175 ROTATIONS Differential Equations of the Rotation Matrices (continue – 3) SOLO Let find the relation between and  B AB     A AB   For any vector let perform the following computationsv            A AB B A B AB BB AB vCvv                   BA B A AB B A AB A A B A AB B A AA AB B A vCCvCCCvC     Since this is true for any vector we havev        A B A AB B A B AB CC     Pre-multiplying by and post-multiplying by we get: A BC B AC       B A B AB A B A AB CC     Rotation Matrix (continue – 14)
  • 176. 176 ROTATIONS Differential Equations of the Rotation Matrices (continue – 4) SOLO Let differentiate the equation 33x A B B A ICC  to obtain       0  dt dC C dt dC CCC dt dC CC dt dC A BB A B AB A BB A A B B A B AB A BB A A B B A   Post-multiplying by we get A BC          A B A AB A B B A B AB A B B AB A B A B CCCCC dt dC     We obtained for the differentiation of the Rotation Matrix                    B AB A B A B A AB A B A BA A B ttCtCttCt dt tdC     Note We can see that            tttt ABBA A AB A BA     End Note Rotation Matrix (continue – 15)
  • 177. 177 ROTATIONS Differential Equations of the Rotation Matrices (continue – 5) SOLO Suppose that we have a third frame of coordinates I (for example inertial) and we have the angular velocity vectors of frames A and B relative to I. We have    B I B IB B I C dt dC       A I A IA A I C dt dC    A I B A B I CCC  dt dC CC dt dC dt dC A IB A A I B A B I        I A A I A IA B A I A B I B IB I A A IB A I A B I B A CCCCCC dt dC CC dt dC dt dC     or From which we get:      A IA B A B A B IB B A CC dt dC     Rotation Matrix (continue – 16)
  • 178. 178 ROTATIONSSOLO From the equation Computation of the Angular Velocity Vector from .AB     nRtC x B A ˆ,33         tCt dt tdC B A B AB B A    we obtain         TB A B AB AB tC dt tdC t   Since the Rotation Matrix is defined also by and          sinˆcos1ˆˆcosˆ, 3333   nnnInRC T x B A  tC B A nˆ we can compute as function of and their derivativesnˆAB  td d   td nd n ˆ ˆ   (this is a long procedure described in the complementary work “Notes on Rotations”, and a simpler derivation will be given later, we give here the final result)      sinˆcos1ˆˆˆ    nnnnAB  Rotation Matrix (continue – 17)
  • 179. 179 ROTATIONSSOLO Computation of and as functions of .AB  td d   td nd n ˆ ˆ   Let pre-multiply the equation by and use T nˆ     sinˆcos1ˆˆˆ    nnnnAB    0ˆˆ,0ˆˆ,1ˆˆ   nnnnnn TTT to obtain     AB TTTT AB T nnnnnnnnn       ˆsinˆˆcos1ˆˆˆˆˆˆ Let pre-multiply the equation by and use nˆ     sinˆcos1ˆˆˆ    nnnnAB           nnInnnnnnn x T ˆˆˆˆˆˆˆ,0ˆˆ 33 to obtain                 sinˆˆcos1ˆsinˆˆcos1ˆˆˆˆˆˆ    nnnnnnnnnnn AB  Let pre-multiply the equation by       sinˆˆcos1ˆˆ    nnnn AB   nˆ               cos1ˆˆsinˆsinˆˆˆcos1ˆˆˆˆ    nnnnnnnnnn AB  Rotation Matrix (continue – 18)
  • 180. 180 ROTATIONS Computation of and as functions of (continue – 1) SOLO AB  td d   td nd n ˆ ˆ   We have two equations:       ABnnnn      ˆsinˆˆcos1ˆ        ABnnnnn      ˆˆcos1ˆˆsinˆ with two unknowns and  nˆ     nn ˆˆ From those equations we get:            sinˆˆcos1ˆsincos1ˆ 22 ABAB nnnn     or           sinˆˆcos1ˆcos1ˆ2 ABAB nnnn     Finally we obtain: AB T n    ˆ      ABnnnn                   2 cotˆˆˆ 2 1 ˆ Rotation Matrix (continue – 19)
  • 181. 181 ROTATIONS Quaternions SOLO The quaternions method was introduced by Hamilton in 1843. It is based on Euler Theorem (1775) that states: The most general displacement of a rigid body with one point fixed is equivalent to a single rotation about some axis through that point. Therefore every rotation is defined by three parameters: • Direction of the rotation axis , defined by two parameters • The angle of rotation , defines the third parameter nˆ  William Rowan Hamilton 1805 - 1865       sinˆcos1ˆˆ1 vnvnnvv   The rotation of around by angle is given by:nˆ v  A B C O   nˆ v  1v  that can be writen        sinˆcos1ˆˆ1 vnvvnnvv   or       sinˆcos1ˆˆcos1 vnvnnvv  
  • 182. 182 ROTATIONS Quaternions (continue – 1) SOLO Computation of the Rotation Matrix We found the Rotation Matrix that describes this rotation from A to B.  ,ˆ33 nRC x B A                    sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ sinˆˆcos1ˆˆˆˆˆ AAAB AAAB AAAB znznnxz ynynnxy xnxnnxx    Axˆ Azˆ Ayˆ Bzˆ Byˆ Bxˆ O nˆ                       A A A B AAA x A B xCnnnIx ˆ 0 0 1 sinˆcos1ˆˆˆ 33                                A A A B AAA x A B yCnnnIy ˆ 0 1 0 sinˆcos1ˆˆˆ 33                                A A A B AAA x A B zCnnnIz ˆ 1 0 0 sinˆcos1ˆˆˆ 33              or from which                  ,ˆsinˆcos1ˆˆ 3333 nRnnnICC x AAA x A B A B 
  • 183. 183 ROTATIONS Quaternions (continue – 2) SOLO Definition of the Quaternions Axˆ Azˆ Ayˆ Bzˆ Byˆ Bxˆ O nˆ     The quaternions (4 parameters) were defined by Hamilton as a generalization of the complex numbers   32100 , qkqjqiqqq     2/cos0 q  nˆ2/sin          zyx nqnqnq 2/sin&2/sin&2/sin 111   where satisfy the relations:kji  ,, 1 kkjjii  kijji   , ijkkj   , jkiik   1 kji  i  j  k  the complex conjugate of is defined asq   32100 * , qkqjqiqqq   
  • 184. 184 ROTATIONS Quaternions (continue – 3) SOLO Product of Quaternions Product of two quaternions andAq Bq      3210321000 ,, BBBBAAAABBAABA qkqjqiqqkqjqiqqqqq         3210321033221100 AAABBBBABABABABA qkqjqiqqkqjqiqqqqqqqqq        122131132332 BABABABABABA qqqqkqqqqjqqqqi   therefore        BAABBABABABBAABA qqqqqqqq    000000 ,,, Let use this expression to find       2 3 2 2 2 1 2 0 222 000 * 00 * 1ˆˆ 2 sin 2 cos,,,, qqqqnnqqqqqqqqq                 The quaternion product can be writen in matrix form as:                                            A A BxBB T BB B B AxAA T AA BA q Iq qq Iq q qq q q           0 330 00 330 00 1 kjikkjjii  kijji   ijkkj   jkiik  
  • 185. 185 ROTATIONS Quaternions (continue – 4) SOLO Rotation Description Using the Quaternions Let compute the expression:                                               AAAAAAAA AAAAA vvqvqvqvvvqqv qvvqvqvqqvq 00 2 000 0000 * , ,,,,0,                                                   A AAAA AAAAAAA vqq vvqqvv vvqvqvqvvv          22,0 2,0 ,0 0 2 0 0 2 0 00 2 0 Using the relations:                                       nnq nnnn q n q ˆsinˆ2/sin2/cos22 ˆˆcos1ˆˆ2/sin22 1 ˆ2/sin 2/cos 0 2 2 0 0          and                     AAAA x AB A B vnnnIvCv   sinˆcos1ˆˆ33  we obtain                         AAABB vqqvqqvqvv    221,0,,0,,0 000 *
  • 186. 186 ROTATIONS Quaternions (continue – 5) SOLO Rotation Description Using the Quaternions (continue – 1) Using the fact that we obtain:          22 033 qIC x B A                                                     0 0 0 0 0 0 2 0 0 0 2 100 010 001 12 13 23 12 13 23 12 13 23 0 qq qq qq qq qq qq qq qq qq q                                          2 2 2 13231 32 2 1 2 321 3121 2 2 2 3 1020 1030 2030 2222 2222 2222 022 202 220 100 010 001 qqqqqq qqqqqq qqqqqq qqqq qqqq qqqq                 2 2 2 110323120 3210 2 1 2 33021 20312130 2 2 2 3 2212222 2222122 2222221 qqqqqqqqqq qqqqqqqqqq qqqqqqqqqq 1 2 3 2 2 2 1 2 0  qqqq                             2 3 2 2 2 1 2 010323120 3210 2 3 2 2 2 1 2 03021 20312130 2 3 2 2 2 1 2 0 22 22 22 qqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqqq C B A
  • 187. 187 ROTATIONS Quaternions (continue – 6) SOLO Rotation as a Multiplication of Two Matrices          22 033 qIC x B A           22 033 2 0 qIq x T               33033 2 0 2 x T x IqIq                  33330330 x T xx IIqIq For any vector we can write         aaaa  or in matrix notation           T x T x T x TT IIaIa    333333 Therefore we have                  33330330 x T xx B A IIqIqC            T xx IqIq   330330  321 3 2 1 012 103 230 012 103 230 qqq q q q qqq qqq qqq qqq qqq qqq                                      
  • 188. 188 ROTATIONS Quaternions (continue – 7) SOLO Rotation as a Multiplication of Two Matrices (continue – 1)                                                                    330 330 012 103 230 321 0123 1032 2301 x T x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C                                                                      330 330 012 103 230 321 0123 1032 2301 x T x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C                                                       T x x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C             330 330 321 012 103 230 3012 2103 1230                                                         T x x B A Iq Iq qqq qqq qqq qqq qqqq qqqq qqqq C             330 330 321 012 103 230 3012 2103 1230
  • 189. 189 ROTATIONS Quaternions (continue – 8) SOLO Relation Between Quaternions and Euler Angles     x A x B x qvqv iq * 2 sin 2 cos                 Rotation around x axis   Ax  1 Ay  1 Az  1 Bz  1 By  1      y A y B y qvqv jq * 2 sin 2 cos                   Ax  1 Ay  1 Az  1 Bz  1 Bx  1  Rotation around y axis     z A z B z qvqv kq * 2 sin 2 cos                   Ax  1 Ay  1 Az  1 Bx  1 By  1 Rotation around z axis
  • 190. 190 ROTATIONS Quaternions (continue – 9) SOLO Description of Successive Rotations Using Quaternions Let describe two consecutive rotations: - First rotation defined by the quaternion                      1 11 1101 ˆ 2 sin, 2 cos, nqq    - Folowed by the second rotation defined by the quaternion                      2 22 2202 ˆ 2 sin, 2 cos, nqq    After the first rotation the quaternion of the vector is transferred to 1 * 1 qvq After the second rotation we obtain      21 * 2121 * 1 * 221 * 1 * 2 qqvqqqqvqqqqvqq  Therefore the quaternion representing those two rotation is:                                                                21 21 1 2 2 1 21 2121 21120210212010220110210 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cos,ˆˆ 2 sin 2 sin 2 cos 2 cos ,,,, nnnnnn qqqqqqqqqq       210 , qqqq    21 2121 0 ˆˆ 2 sin 2 sin 2 cos 2 cos 2 cos nnq                                 21 21 1 2 2 1 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cosˆ 2 sin nnnnn                                  
  • 191. 191 ROTATIONS Quaternions (continue – 10) SOLO Description of Successive Rotations Using Quaternions (continue – 1)    210 , qqqq    21 2121 0 ˆˆ 2 sin 2 sin 2 cos 2 cos 2 cos nnq                                 21 21 1 2 2 1 ˆˆ 2 sin 2 sinˆ 2 cosˆ 2 cosˆ 2 sin nnnnn                                   Two consecutive rotations, followed by , are given by:1q 2q From those equations we can see that: 0ˆˆˆˆˆˆˆˆˆˆ 21212112211221   nnnnnnnnnnifonlyandifqqqq The rotations are commutative if and only if are collinear.21 ˆ&ˆ nn In matrix form those two rotations are given by: First Rotation:         111111331133 sinˆcos1ˆˆcosˆ,   nnnInR T xx Second Rotation:         222222332233 sinˆcos1ˆˆcosˆ,   nnnInR T xx Total Rotation:              sinˆcos1ˆˆcosˆ,ˆ,ˆ, 331133223333  nnnInRnRnR T xxxx
  • 192. 192 ROTATIONS Quaternions (continue – 11) SOLO Description of Successive Rotations Using Quaternions (continue – 2) Let find the quaternion that describes the Euler Rotations through the angles respectively. Let write the rotations according to their order 123   ,,                                                        2 sin 2 cos 2 sin 2 cos 2 sin 2 cos  ijkqqqq xyz B A                                                                           2 sin 2 sin 2 cos 2 sin 2 cos 2 sin 2 cos 2 cos 2 sin 2 cos  kjik                                             2 sin 2 sin 2 sin 2 cos 2 cos 2 cos                                             2 cos 2 sin 2 sin 2 sin 2 cos 2 cos  i                                             2 cos 2 sin 2 cos 2 sin 2 cos 2 sin  j                                             2 sin 2 sin 2 cos 2 cos 2 cos 2 sin  k 
  • 193. 193 ROTATIONS Quaternions (continue – 12) SOLO Differential Equation of the Quaternions Let define      ,0qtq B A  - the quaternion that defines the position of B frame relative to frame A at time t.    tqqttq B A    ,00 - the quaternion that defines the position of B frame relative to frame A at time t+Δt.                        t B A ntq ˆ 2 sin, 2 cos  - the quaternion that defines the position of B frame at time t+Δt relative to frame B at time t. We have the relation:      tqtqttq B A B A B A  or                                            ,,0,1,,,,,ˆ 2 sin, 2 cos 00000000 * qqqqqqqqttqtqntq B A B At B A therefore                         tnqq ˆ 2 sin,1 2 cos,, 00                             tnqq ˆ 2 sin,1 2 cos,, 00    or
  • 194. 194 ROTATIONS Quaternions (continue – 13) SOLO Differential Equation of the Quaternions (continue – 1)                          tnqq ˆ 2 sin,1 2 cos,, 00    Let divide both sides by and take the limit .0 tt                                                                                        tBttB t ntqnqn tt tq td d tt q ˆ 2 1 ,0ˆ 2 1 ,0,ˆ 2 2 sin 2 1 , 2 1 2 cos 2 1 ,lim 0 0 0           But is the instant angular velocity vector of frame B relative to frame A.tn ˆ     t B AB nt   ˆ            ttn B AB B ABt     ,0ˆ,0 So we can write        ttqtq td d B AB B A B A   2 1 This is the Differential equation of the quaternion that defines the position of B relative to A, at the time t as a function of the angular velocity vector of frame B relative to frame A, .  tq B A    t B AB 
  • 195. 195 ROTATIONS Quaternions (continue – 14) SOLO Differential Equation of the Quaternions (continue – 2) Developing this equation, we get        ttqtq td d B AB B A B A   2 1               B AB B AB B AB B AB qtq dt d dt dq            00 0 , 2 1 ,0, 2 1 , from which  B AB dt dq    2 10      B AB B ABq dt d       0 2 1 or in matrix form        t Iq q dt d B AB x T                                   330 0 2 1
  • 196. 196 ROTATIONS Quaternions (continue – 15) SOLO Differential Equation of the Quaternions (continue – 3)                                                   zBAB yBAB xBAB qqq qqq qqq qqq q q q q dt d    012 103 230 321 3 2 1 0         t Iq q dt d B AB x T                                   330 0 2 1 B AAB xBAByBABzBAB xBABzBAByBAB yBABzBABxBAB zBAByBABxBAB q q q q q q q q q dt d                                                              2 1 0 0 0 0 2 1 3 2 1 0 3 2 1 0            After rearranging or  zBAByBABxBAB qqq dt dq    321 0 2 1  zBAByBABxBAB qqq dt dq    230 1 2 1  zBAByBABxBAB qqq dt dq    103 2 2 1  zBAByBABxBAB qqq dt dq    012 3 2 1
  • 197. 197 ROTATIONS Quaternions (continue – 16) SOLO Pre-multiply the equation Computation of as a Function of the Quaternion and its Derivatives    t B AB         t Iq q dt d B AB x T                                   330 0 2 1 by          330 xIq                                                t Iq Iq q Iq B AB x T xx                   330 330 0 330 2 1                        ttIIq tIq B BA B BAx TT x T B BAx T         2 1 2 1 2 1 3333 2 0 33 2 0 Therefore                                 0 3302 q Iqt x B AB
  • 198. 198 ROTATIONS Quaternions (continue – 17) SOLO Computation of as a Function of the Quaternion and its Derivatives (continue – 1) But and are related. Differentiating the equation    t B AB  we obtain  0q    1 2 0   T q                                                 3300 0 330 22 xx B AB Iqq q Iqt                          0 033 2 0 330 0 2 1 2 q qIq Iq q T x x T From the equation     TT q qqq 0 000 1 0 we obtain             T x B AB qIq q 033 2 0 0 2
  • 199. 199 ROTATIONS Quaternions (continue – 18) SOLO Computation of as a Function of , and their Derivatives   t B AB   nˆ Differentiate the quaternion                      nqq ˆ 2 sin, 2 cos,0    to obtain                                 nnqq ˆ 2 sinˆ 2 cos 2 , 2 sin 2 ,0     Substitute this in the equation                                 0 3302 q Iqt x B AB                                                                nn nIn x ˆ 2 sinˆ 2 cos 2 2 sin 2 ˆ 2 sin 2 cosˆ 2 sin2 33                                                        nnnnnnn ˆˆ 2 sin2ˆˆ 2 cos 2 sinˆ 2 cos 2 sin2ˆ 2 cosˆ 2 sin 222               nnnnAB ˆˆcos1ˆsinˆ   Finally we obtain We recovered a result obtained before.
  • 200. 200 ROTATIONS Quaternions (continue – 18) SOLO Differential Equation of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I The relations between the components of a vector in the frames A, B and I arev    AB A IA I B A IB I B vCvCCvCv   Using quaternions the same relations are given by   B A A I IA I B A B I IB I B qqvqqqvqv ***  Therefore B A A I B I qqq  B I A I B A qqq *  Let perform the following calculations B A A I B A A I B I q dt d qqq dt d q dt d  & B IB B I B I qq dt d   2 1  A IA A I A I qq dt d   2 1 and use     B A A I B A A IA A I B IB B I q dt d qqqq    2 1 2 1     B A A IA A I A I B IB B I A I B A qqqqqq dt d     1 ** 2 1 2 1 to obtain     B A A IA B IB B A B A qqq dt d    2 1 2 1
  • 201. 201 ROTATIONS Quaternions (continue – 19) SOLO Differential Equatio of the Quaternion Between Two Frames A and B Using the Angular Velocities of a Third Frame I (continue – 1) Using the relations ABIAIB     and     B A A IA B A B IA qq    *     B A A IA B IA B A qq   we have                  0 2 1 2 1 2 1 2 1 2 1 B A A IA B IA B A B AB B A B A A IA B IA B AB B A B A qqqqqq dt d    from which     B A A AB B AB B A B A qqq dt d    2 1 2 1 Since BAAB     we get         B A A BA B BA B A B A A AB B AB B A B A qqqqq dt d    2 1 2 1 2 1 2 1 From we get1 *  B A A B B A B A qqqq A B B A qq  * Therefore              B A A B B A A B q dt d qqq dt d 0 *B A B A A B A B qq dt d qq dt d             Table of Contents
  • 202. 202 SOLO Laplace Fields Vector Analysis A vector field is said to be a Laplace Field if rAA     0 rA  In this case we have and       022 00 2         AAAAA   0 rA  Harmonic Functions A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions Pierre-Simon Laplace 1749-1827 022             SS dS n dS n 2  0 0 2 1     V GAUSS SS dvdSdS n   Proof: 1 0   S dS n  Proof:   0 0 2 0 2                     dvdS nnS       n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   2nd Green’s Identity:
  • 203. 203 SOLO Vector Analysis Harmonic Functions (continue 1) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 1)  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   3 A function φ harmonic in a volume V can be expressed in terms of the function and its normal derivative on the surface S bounding V. Proof: Use the solution of the Poisson (Laplace) Equation: 02                             S SFSF F dS rrnnrr T r 11 4      where VoutsidendSndS SonF VinF T        11 2 1 1
  • 204. 204 SOLO Vector Analysis Harmonic Functions (continue 2) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 2) RS dSn  1 V Fr  Sr  F SF rrR   4 If the surface S is a sphere SR of radius R with center at then    RS RF dS R r    2 4 1 Fr  Proof: 0 11         SS SF dS nR dS nrr  1 2 11 Rrrn RS SF            Therefore:                            RS R S SFSF F dS R dS rrnnrr T r       2 4 111 4  3 5 If φ is harmonic in a volume V bounded by the surface S and if φ = c = constant at every point on S, then φ = c at every point of V. Proof:     cdc rr dS cdS rrnnrr r S SFS rr SFcSF F SF                                                          4 2 /1 0 4 1 4 111 4 1 2     3
  • 205. 205 SOLO Vector Analysis Harmonic Functions (continue 3) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 3) 6 A non-constant function φ harmonic in a region V can have neither a maximum nor a minimum in V. Proof: For any point inside the region V we can choose an infinitesimal sphere δS centered at this point for which         n n dvdSdS n VSS 100 0      Since can not be either positive or negative inside the region V, the maximum or minimum of the potential φ can occur only at boundary of the region. n  S dSn  1 V Fr  Sr  SF rrr   SF
  • 206. 206 SOLO Vector Analysis Harmonic Functions (continue 4) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 4) 7 If φ is harmonic in a region V bounded by a surface S and ∂ φ/∂ n = 0 at every point of S, then φ = constant at every point of V. S dSn  1 V Fr  Sr  FSF rrr   0   S n  Proof:        SVV dSdvdv  2 Use: 0& 2               SV dSdv  0 2    0 0 2      SV dS n dv   Therefore: VinconstVin .0   1st Green’s Identity:
  • 207. 207 SOLO Vector Analysis Harmonic Functions (continue 5) A continuous function φ with continuous first and second partial derivatives is said to be harmonic if it satisfies Laplace’s Equation 02   Properties of Harmonic Functions (continue – 4) 8 If φ1 and φ2 are two solutions of Laplace’s equation in a volume V whose normal derivatives take the same value ∂ φ1/∂ n = ∂ φ2/∂ n on the surface S bounding V, then φ1 and φ2 can differ only by a constant. S dSn  1 V Fr  Sr  FSF rrr   0   S n  Proof: Define: 0: 2 2 1 22 21   021          SSS nnn  From it follows that φ is a constant.7 Table of Contents
  • 208. 208 SOLO Fundamental Theorem of Vector Analysis for a Bounded Region V (Helmholtz’s Theorem) Vector Analysis Hermann Ludwig Ferdinand von Helmholtz 1821 - 1894 Let be a continuous vector field with continuous divergence and curl, in a region V bounded by a surface S. Then has a unique representation as sum of a potential field and a solenoidal field , i.e.  rAA   1 A    2 A A           FFFFF rArArA 21  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr                  S FS S V FS SS V FS S F dS rr rA dv rr rA dv rr rA          4 1 4 1 4 1 :                        S FS S V FS SS V FS SF dS rr rA dv rr rA dv rr rA         4 1 4 11 4 1 :                                         S FS S V FS SS F S FS S V FS SS FF dS rr rA dv rr rA dS rr rA dv rr rA rA            4 1 4 1 4 1 4 1 Therefore
  • 209. 209 SOLO Proof of the Fundamental Theorem of Vector Analysis for a Bounded Region V Vector Analysis Let use the fact that (see GREEN’s FUNCTION):  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr     zzyyxxr SSSS 111    zzyyxxr FFFF 111  where We can write:           z z y y x x SSS S 111 Sr  We define the operator that accts only on . The operator acts only on .           z z y y x x FFF F 111 Fr   SF FS F rr rr              4 12                     1 0 00 dxx x x x                                 V FS SF V FS FS V SFSF dv rr rAdv rr rAdvrrrArA     1 4 11 4 1 22   Using the identity we obtain:  2 FFFFFF                            V FS SFF V FS S FFF dv rr rAdv rr rA rA      1 4 1 4 1 
  • 210. 210 SOLO Vector Analysis Let develop first the divergence expression:  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr                                                                    S FS S V SS FS V FS S S V SS FS V FS SS V FS FS V FS SF dS rr rA dvrA rr dv rr rA dvrA rr dv rr rAdv rr rAdv rr rA                  4 11 4 1 4 11 4 1 1 4 11 4 11 4 1 Define:                       S FS S V SS FSV FS SF dS rr rA dvrA rr dv rr rA        4 11 4 11 4 1 :                           V FS SFF V FS S FFF dv rr rAdv rr rA rA      1 4 1 4 1  Proof of the Fundamental Theorem of Vector Analysis for a Bounded Region V (continue – 1)
  • 211. 211 SOLO Vector Analysis Let develop now the rotor expression:  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Define:                S FS S V FS SS V FS S F dS rr rA dv rr rA dv rr rA          4 1 4 1 4 1 :                                                V FS S S V FS SS V FS SS V FS FS V FS S F dv rr rA dv rr rA dv rr rAdv rr rAdv rr rA             4 1 4 1 1 4 11 4 1 4 1 but                                                  S FS S S FS S S FS S V FS S S tconsC V FS S S dS rr rA CdS rr rA C dS rr rA Cdv rr rA Cdv rr rA C                   4 1 4 1 4 1 4 1 4 1                   S FS S GAUSS V FS S S dS rr rA dv rr rA      4 1 4 1 5                           V FS SFF V FS S FFF dv rr rAdv rr rA rA      1 4 1 4 1  Gauss 5 Proof of the Fundamental Theorem of Vector Analysis for a Bounded Region V (continue – 2)
  • 212. 212 SOLO Vector Analysis We found:  n i iSS 1  iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Therefore:                S FS S V FS SS V FS S F dS rr rA dv rr rA dv rr rA          4 1 4 1 4 1 :                           V FS SFF V FS S FFF dv rr rAdv rr rA rA      1 4 1 4 1                        S FS S V SS FSV FS SF dS rr rA dvrA rr dv rr rA        4 11 4 11 4 1 :    FFF rA   q.e.d. Proof of the Fundamental Theorem of Vector Analysis for a Bounded Region V (continue – 3) Table of Contents
  • 213. 213 SOLO Fundamental Theorem of Vector Analysis for an Unbounded Region (Helmholtz’s Theorem) Vector Analysis For a Bounded region V we found: Hermann Ludwig Ferdinand von Helmholtz 1821 - 1894 Let be a continuous vector field with continuous divergence and curl, such that falls off at infinity like 1/r 1+ε while and fall off at infinity like 1/r 2+ε where ε > 0. Then has a unique representation (to within constant vectors) at sum of a potential field and a solenoidal field , i.e.  rAA   A  A   A   A   rAA  11   rAA  22        0&0 4 1 4 1                                 UU Udv rr rA dv rr rA rA FF V FS SS F V FS SS FF                                          S FS S V FS SS F S FS S V FS SS FF dS rr rA dv rr rA dS rr rA dv rr rA rA            4 1 4 1 4 1 4 1 3/4&4 32 RVRSRrr FS    For an Unbounded region V: The surface integrals are finite only if falls off at infinity like 1/r 1+ε where ε > 0. rA  The volume integrals are finite only if and fall off at infinity like 1/r 2+ε.A   A  
  • 214. 214 SOLO Fundamental Theorem of Vector Analysis for an Unbounded Region (Helmholtz’s Theorem) Vector Analysis     0,0 11  rArA  Therefore Hermann Ludwig Ferdinand von Helmholtz 1821 - 1894 Let be a continuous vector field with continuous divergence and curl, such that falls off at infinity like 1/r 1+ε while and fall off at infinity like 1/r 2+ε where ε > 0. Then has a unique representation (to within constant vectors) at sum of a potential field and a solenoidal field , i.e.  rAA   A  A   A   A   rAA  11   rAA  22        0&0 4 1 4 1                                 UU Udv rr rA dv rr rA rA FF V FS SS F V FS SS FF                                          S FS S V FS SS F S FS S V FS SS FF dS rr rA dv rr rA dS rr rA dv rr rA rA            4 1 4 1 4 1 4 1 Table of Contents
  • 215. 215 REYNOLDS’ TRANSPORT THEOREM This is a part of the Presentations “FLUID DYNAMICS” v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or,  - any system of coordinatesOxyz - any continuous and differentiable functions in    trtr OO ,,, ,,    tandrO,   trO ,,   - flow density at point and time t Or,  SOLO - mass flow through the element .mdsdVS    , sd  - any control volume, changing shape, bounded by a closed surface S(t)v (t) - flow velocity, relative to O, at point and time t trV OOflow ,,,  Or,  - position and velocity, relative to O, of an element of surface, part of the control surface S(t). OSOS Vr ,, ,  - area of the opening i, in the control surface S(t).iopenS - gradient operator in O frame.O, - flow relative to the opening i, in the control surface S(t).OSiOflowSi VVV ,,,   - differential of any vector , in O frame. O td d     FLUID DYNAMICS FLUID DYNAMICS MATHEMATICS SOLO HERMELIN Updated: 5.03.07
  • 216. 216 Start with LEIBNIZ THEOREM from CALCULUS:        ChangeBoundariesthetodueChange tb ta tb ta td tad ttaf td tbd ttbfdx t txf dxtxf td d LEIBNITZ         )),(()),(( ),( ),(:: )( )( )( )(   and generalized it for a 3 dimensional vector space on a volume v(t) bounded by the surface S(t). Using LEIBNIZ THEOREM followed by GAUSS THEOREM (GAUSS 4):                   tv OSOOOSGAUSS OpointtoRelative dsofMovement thetodueChange tS OS tv O LEIBNITZ Otv vdVV t GAUSS sdVvd t vd td d ,,,,)4( )( ,               This is REYNOLDS’ TRANSPORT THEOREM OSBORNE REYNOLDS 1842-1912 SOLO GOTTFRIED WILHELM von LEIBNIZ 1646-1716 REYNOLDS’ TRANSPORT THEOREM v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or,  FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE)
  • 217. 217 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION               )( ,,,,)4( , )()()( tv OSOOOS O GAUSS OS tStv O LEIBNITZ O tv vdVV t GAUSS sdVvd t vd td d                        )( , , )4( , )()()( tv k kOS i k i kOS i GAUSS kkOS tS i tv i LEIBNITZ tv i vd x V x V t GAUSS sdVvd t vd td d            SOLO v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or, 
  • 218. 218 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION O OOS td Rd uV    ,, CASE 1 (CONTROL VOLUME vF ATTACHED TO THE FLUID) kkOS uV ,               )( ,,,)4( , )()()( tv OOO O GAUSS O tStv OO tv F FFF vduu t GAUSS sduvd t vd td d                         )( )4( )()()( tv k k I k I k I GAUSS kK tS I tv I tv I F FFF vd x u x u t GAUSS sduvd t vd td d            SOLO vF (t) SF(t) O x y z sd  OSV ,   0,,,  OSOflowS VVV  OSR , OR,  md OSV ,  OflowV , 
  • 219. 219 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION 1&,  kkOS uV1&,  uV OS  CASE 2 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )1   )( ,, )( , )( )( tv OO tS O tv F FFF vdusduvd td d td tvd    )()()( )( tv k k k tS k tv F FFF dv x u dsudv td d td tvd                   td tvd tv u F F tv OO F )( )( 1 lim0)( ,,                  td tvd tvx u F F tv k k F )( )( 1 lim0)(  EULER 1755 SOLO vF (t) SF(t) O x y z sd  OSV ,   0,,,  OSOflowS VVV  OSR , OR,  md OSV ,  OflowV , 
  • 220. 220 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CASE 3 (CONTROL VOLUME vF ATTACHED TO THE FLUID AND )   &, kkOS uV  &, uV OS    or, since this is true for any attached volume vF(t)             )( ,, )( , )( )( )( 0 tv OO tS O tv tv F FF F vdu t sduvd t vd td d td tmd                      )( )()( )( )( 0 tv k k tS kk tv tv F FF F vdu xt sduvd t dv td d td tmd          Because the Control Volume vF is attached to the fluid and they are not sources or sinks, the mass is constant.   OOOOOO uu t u t ,,,,,, 0           k k k k k x u x u t u xt               0 SOLO vF (t) SF(t) O x y z sd  OSV ,   0,,,  OSOflowS VVV  OSR , OR,  md OSV ,  OflowV , 
  • 221. 221 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,  OSV Define   .... VC OO VC vd t vd td d        .... VC i VC i vd t vd td d                 r t r t r t, , ,        i k k i kx t x t x t, , ,              )( , )()( tS OS tv OO tv sdV vd tt vd td d            k tS kOSi tv i i tv i sdV vd tt vd td d FF           )( , )()(        We have but    OOOO u t u t ,,,, 0            k k iik k u xt u xt             0 CASE 5              r t r t r t, , , SOLO
  • 222. 222 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION We have                                                   )( ,, )( 4 . )( , )( ,,,,,, )( , )( ,, )( tS OOS tv O MDG DerMat GAUSS tS OS tv OOOOOO O tS OS tv OO OO tv sduVvd tD D sdV vduuu t sdV vdu t vd td d                                                                        )( , )( 4 . )( , )( )( , )()( tS kkkOSi tv i MDG DerMat GAUSS tS kkOSi tv k k i k i k k i k i tS kkOSi tv k k i i tv i sduVvd tD D sdV vd x u x u x u t sdV vd x u t vd td d                      CASE 5              r t r t r t, , , SOLO v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or, 
  • 223. 223 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION REYNOLDS 1             )( ,, )( )( tS OOS tv O O tv sduVvd tD D vd td d                   )( , )( )( tS kkkOSi tv i tv i sduVvd tD D dv td d    REYNOLDS 2               )( )( ,, )( tv O tS OSO O tv vd tD D sdVuvd td d                    )( )( , )( tv i tS kkOSki tv i vd tD D sdVuvd td d    CASE 5              r t r t r t, , , SOLO v (t) S(t) O x y z SflowV ,  sd  OSV ,   OSOflowSflow VVV ,,,   OSr ,  md OSV ,  OflowV ,  Or, 
  • 224. 224 FLUID DYNAMICS 1. MATHEMATICAL NOTATIONS (CONTINUE) 1.11 REYNOLDS’ TRANSPORT THEOREM (CONTINUE) VECTOR NOTATION CARTESIAN TENSOR NOTATION REYNOLDS 3 CASE 1 (CONTROL VOLUME ATTACHED TO THE FLUID vF(t) ) kkOS uV ,   )()( tv OO tv FF vd tD D vd td d        )()( tv i tv i FF vd tD D vd td d    SOLO O OOS td Rd uV    ,,              r t r t r t, , , vF (t) SF(t) O x y z sd  OSV ,   0,,,  OSOflowS VVV  OSR , OR,  md OSV ,  OflowV ,  CASE 4 (CONTROL VOLUME WITH FIXED SHAPE C.V. )0,  OSV REYNOLDS 4            .. , .. .. SC O O VC VC O sduvd td d vd tD D                 .... .. SC kki VC i VC i sduvd td d vd tD D    Table of Contents
  • 225. 225 Poisson’s Non-homogeneous Differential EquationSOLO The Poisson’s Non-homogeneous Differential Equation for the Static Electric Scalar Potential Ve is: We want to find the Electric Scalar Potential Ve at the point F (field) due to all the sources (S) in the volume V, including its boundaries . n i iSS 1     SFSeS rrrV    1 , 2  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V Therefore is the vector from S to F.SF rrr     zzyyxxr 111    zzyyxxr SSSS 111    zzyyxxr FFFF 111  Let define the operator that acts only on the source coordinate .           z z y y x x SSS S 111 Sr  Note: See development in the Power Point Presentation Table of Contents
  • 226. 226 Poisson’s Non-homogeneous Differential Equation SOLO To find the solution we used the following: • GREEN’s IDENTITY       S ee V ee dSGVVGdvGVVG 22 • GREEN’s FUNCTION This Green’s Function is a particlar solution of the following Poisson’s Non-homogeneous Differential Equation:       0;; 1 ; 2    FSSFS SF FS rrrr rr rrG       FSFSS rrrrG   4;2 Siméon Denis Poisson 1781-1840 GEORGE GREEN 1793-1841
  • 227. 227 Poisson’s Non-homogeneous Differential EquationSOLO • GREEN’s IDENTITY       S ee V ee dSGVVGdvGVVG 22 Let start from the Gauss’ Identity    SV dSAdvA  Karl Friederich Gauss 1777-1855 where is any vector field (function of position and time) continuous and differentiable in the volume V. Let define . A  eVGA     eee VGVGVGA 2   Then        S e V ee V e dSVGdvVGVGdvVG 2 If we interchange G with Ve we obtain        S e V ee V e dSGVdvGVVGdvGV 2 Subtracting the second equation from the first we obtain First Green’s Identity Second Green’s Identity We have
  • 228. 228 SOLO • GREEN’s FUNCTION Define the Green’s Function is a particlar solution of the following Poisson’s Non-homogeneous Differential Equation: where δ (x) is the Dirac function                     1 0 00 dxx x x x   Let use the Fourier Transformation to write                                                                             3 3 3 exp 2 1 exp 2 1 exp 2 1 exp 2 1 exp 2 1 dkrrkj dkdkdkzzkyykxxkj dkzzjkdkyyjkdkxxjk zzyyxxrr SF zyxSFzSFySFx zSFzySFyxSFx SFSFSFSF       zyx zyx dkdkdkdk zkykxkk    3 111  where    FSFSS rrrrG   4;2 Paul Adrien Maurice Dirac 1902 - 1984 Poisson’s Non-homogeneous Differential Equation
  • 229. 229 SOLO • GREEN’s FUNCTION (continue – 1) Let use the Fourier Transformation to write Hence or         SFFS rrkjkgdkrrG  exp; 3            SFSFS rrkjdkrrkjkgdk  exp 2 4 exp 3 3 32              SFSFS rrkjdkrrkjkgdk  exp 2 4 exp 3 3 23   Poisson’s Non-homogeneous Differential Equation Jean Baptiste Joseph Fourier 1768 - 1830
  • 230. 230 SOLO • GREEN’s FUNCTION (continue – 2) Let compute: Therefore: Because this is true for all k, we obtain            SFzSFySFxSSSFS zzkyykxxkjrrkj  expexp2                       SFzSFySFxzyxS zzkyykxxkjzjkyjkxjk exp111        SFzSFySFxSzyx zzkyykxxkjzjkyjkxjk         exp111             SFSFSFS rrkjkrrkjkjkjrrkjkj   expexpexp 2            SFSF rrkjdkrrkjkkgdk  exp 2 4 exp 3 3 23            22 1 2 1 k kg   Poisson’s Non-homogeneous Differential Equation
  • 231. 231 SOLO • GREEN’s FUNCTION (continue – 3) Let use spherical coordinates relative to vector:r                  rr r r kk kk kk z y x z y x 0 0 cos sinsin cossin     dk sin dk dk     dddkkdk sin23    r x y z         SFFS rrkjkgdkrrG  exp; 3          22 1 2 1 k kg   Poisson’s Non-homogeneous Differential Equation
  • 232. 232 SOLO • GREEN’s FUNCTION (continue – 4)        2 3 2 exp 2 1 ; k rkj dkrrG FS           0 0 2 0 2 22 sin cosexp 2 1      dkddk k jkr       0 0 2 coscosexp2 2 1    dkdjkr                      00 0 2 expexp2cosexp1 dk kj jkrjkr r dk jkr jkr        rr dk k kr r 1 2 2sin2 0      Poisson’s Non-homogeneous Differential Equation   2 sin 0    dk k kr where we used (see next slide)   SF FS rrr rrG     11 ;Therefore                 rr r r kk kk kk z y x z y x 0 0 cos sinsin cossin     dk sin dk dk     dddkkdk sin23    r x y z
  • 233. 233 SOLO • GREEN’s FUNCTION (continue – 5) Poisson’s Non-homogeneous Differential Equation     0 sin dk k kr Let compute: x y R  A B C D E F G H Rx Rx  For this use the integral: 0ABCDEFGHA zi dz z e Since z = 0 is outside the region of integration 0    BCDEF ziR xi GHA zi R xi ABCDEFGHA zi dz z e dx x e dz z e dx x e dz z e                  00 0000 sin 2 sin 2 sin lim2limlimlim dk k rk idx x x idx x x idx x ee dx x e dx x e R R R xixi R R xi RR xi R                       idideidei e e dz z e i ii eii i eiez GHA zi      00 1 0 0 00 limlimlim    01 2 2 0 /2 /2sin 0 sin 00        R RRReRii i eRieRz BCDEF zi e R dedededeRi eR e dz z e i ii             Therefore: 0 sin 2 0    idk k rk idz z e ABCDEFGHA zi   2 sin 0    dk k kr
  • 234. 234 Poisson’s Non-homogeneous Differential EquationSOLO • GREEN’s FUNCTION (continue – 6) a Green’s Function for the Poisson’s Non- homogeneous Differential Equation   SF FS rr rrG     1 ; Hence This solution is not unique since we can add any function that satisfies the Laplace’s Equation   0;2  FSS rr  Therefore we have the following Green’s Function       0;; 1 ; 2    FSSFS SF FS rrrr rr rrG    Pierre-Simon Laplace 1749-1827    FSFSS rrrrG   4;2
  • 235. 235 Poisson’s Non-homogeneous Differential EquationSOLO Let return to the Poisson’s Non-homogeneous Differential Equation (1812) for the Electric Scalar Potential Ve is: We want to find the Electric Scalar Potential Ve at the point F (field) due to all the sources (S) in the volume V, including its boundaries  n i iSS 1  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V    SFSeS rrrV    1 , 2  Siméon Denis Poisson 1781-1840
  • 236. 236 Poisson’s Non-homogeneous Differential EquationSOLO Let define the operator that acts only on the source coordinate .Sr            z z y y x x SSS S 111 is the vector from S to F.SF rrr     zzyyxxr 111    zzyyxxr SSSS 111    zzyyxxr FFFF 111  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   0       r r r rr rr rr rr rrr SF SF SF FS SFSS         30  SSSFSS rrrr    0 1 2  r r r rr r dr dG r dr dG rG SS        003 1311 34333 2                    r r r r r r r r r rr r rGrG SSSSSS        SFSeS rrrV    1 , 2  Since is no defined at r = 0 we define the volume V’ as the volume V minus a small sphere of radius and surface around the point F, when F is inside V, or a semi-sphere of radius and surface around the point F, when F is on the boundary of V.  rG 00 r 00 r 2 04 rSF  2 02 rSF 
  • 237. 237 Poisson’s Non-homogeneous Differential EquationSOLO let compute                  '' '0 2 '' 22 , 1 , V SFS V Vin Se V S FS V SeeS dVrrrGdvGVdv r rrGdvGVVG                               FF S SeeS S SeeS dSn r VV r dSGVVG 11                       drn r VV r SeeS r 2 0 11 lim          0 2 0 2 20 0 2 0 0 0 limlimlimlim                     drnVdrn r n VdnVrdnVr Se r e r eS r eS r             VoutsideF SonF VinF rVdrV Fee rr FS 0 2 4 lim     iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Using the Green’s Identity       S ee V ee dSGVVGdvGVVG 22
  • 238. 238 Poisson’s Non-homogeneous Differential EquationSOLO We obtain                                   S e e V SSF dSndS n n S SeeS V SSFFe dS n G V n V G T dvrrrG T dSGVVG T dvrrrG T rV S       4 , 4 4 , 4 1 1   where VoutsidendSndS SonF VinF T        11 2 1 1 Note If F is outside V from the Green’s Second Identity we obtain End Note     VoutsidendSndSdS n G V n V GdSGVVGdvGVVG S e e S SeeS V SeeS               11 22 iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr     SF FS rr rrG     1 ;
  • 239. 239        S D e S DSeFe dS n G V T dSGV T rV  44  where VoutsidendSndS SonF VinF T        11 2 1 1 Poisson’s Non-homogeneous Differential EquationSOLO BOUNDARY CONDITIONS The General Green Function that is a class of bi-position function, and contains an arbitrary harmonic function (solution of the Laplace’s Equation)       0;; 1 ; 2    FSSFS SF FS rrrr rr rrG    Let consider the following two simple cases (Dirichlet and Neumann Problems): 1. Dirichlet Problem Johann Peter Gustav Lejeune Dirichlet 1805-1859 The potential is defined at the boundary S of the volume V.     n i iFe SSongivenrV 1  In this case Let choose such that FS rr  ;   SrrrGG SSFSDirichlet   0; iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
  • 240. 240 Poisson’s Non-homogeneous Differential EquationSOLO BOUNDARY CONDITIONS (continues – 1) 2. Neumann Problem The potential derivative is defined at the boundary S of the volume V. In this case   S S nSSongivennVr n V n i i S eSF e         1&1 1   Let choose such that FS rr  ;     SrnrrG n rrG G S S FSS S FS Neumann        01; ; where VoutsidendSndS SonF VinF T        11 2 1 1        S e N S eSNFe dS n V G T dSVG T rV  44  Franz Neumann 1798-1895 iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
  • 241. 241 Poisson’s Non-homogeneous Differential EquationSOLO Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions Suppose that we have a solution Ve that satisfies the Laplace Homogeneous Differential Equation:   0, 2  FSeS rrV  in the volume V, including its boundaries . n i iSS 1  Suppose also that Dirichlet or Neumann conditions or a combination of those, are specified. In this case the solution is unique (up to an additive constant). Proof Suppose that thee exist two solutions and , and define rVe1  rVe 2      rVrVr ee 21   We have           0 1 2 2 1 1 22     SS r e r e rVrVr    
  • 242. 242 Poisson’s Non-homogeneous Differential EquationSOLO Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions If Dirichlet conditions are satisfied: Proof (continue) Let use Green’s First Identity (with G = Φ) (continue – 1)              n i i rVrV FeFeF SSonrVrVr FeFe 1 21 0 21    If Neumann conditions are satisfied:              n i i r n V r n V F e F e F SSonr n V r n V r n F e F e 1 21 0 21                         SSVV dS n dSGdvdv 2 We have   VinconstVVVindS n dv ee Dirichlet Neumann SV      21 00 End of Proof
  • 243. 243 Poisson’s Non-homogeneous Differential EquationSOLO In the same way the solution of the Poisson’s Non-homogeneous Differential Equation for the Vector Potential is: F rA                                                                          S SF S S SFV SF SS dSndS n n S SF SSSS SFV SF SS F dS rrn rA n rA rr T dv rr rAT dS rr rArA rr T dv rr rAT rA S          11 44 11 44 2 1 1 2   where VoutsidendSndS SonF VinF T        11 2 1 1 iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Table of Contents
  • 244. 244 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION The Helmholtz Non-homogeneous Differential Equation for the Electric Scalar Potential Ve is:      trtrV tv trV eee , 1 , 1 , 2 2 2 2        We want to find the Electric Scalar Potential Ve at the point F (field) due to all the sources (S) in the volume V, including its boundaries . n i iSS 1  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V Therefore is the vector from S to F.SF rrr     zzyyxxr 111    zzyyxxr SSSS 111    zzyyxxr FFFF 111  Let define the operator that acts only on the source coordinate .           z z y y x x SSS S 111 Sr  This is a part of the presentation “Electromagnetics” SOLO HERMELIN ELECTROMAGNETICS
  • 245. 245 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION To find the solution we need to prove the following: • GREEN’s IDENTITY       S ee V ee dSGVVGdVGVVG 22 • GREEN’s FUNCTION   FS FS FS rr v rr tt trtrG                ' ',;,  This Green’s Function is a particular solution of the following Helmholtz Non-homogeneous Differential Equation:        '4',;, 1 ',;, 2 2 2 2 ttrrtrtrG tv trtrG SFFSFSS       (continue – 1) iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V
  • 246. 246 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • Scalar Green’s Identities       S ee V ee dSGVVGdVGVVG 22 (continue – 2) Let start from the Gauss’ Divergence Theorem    SV dSAdVA  Karl Friederich Gauss 1777-1855 where is any vector field (function of position and time) continuous and differentiable in the volume V. Let define . A  eVGA     eee VGVGVGA 2   Then        S e Gauss V ee V e dSVGdVVGVGdVVG 2 If we interchange G with Ve we obtain        S e Gauss V ee V e dSGVdVGVVGdVGV 2 Subtracting the second equation from the first we obtain First Green’s Identity Second Green’s Identity We have GEORGE GREEN 1793-1841
  • 247. 247 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION Define the Green’s Function as a particular solution of the following Helmholtz Non-homogeneous Differential Equation:        '4',;, 1 ',;, 2 2 2 2 ttrrtrtrG tv trtrG SFFSFSS       (continue – 3) where δ (x) is the Dirac function                     1 0 00 dxx x x x   Let use the Fourier Transformation to write                                                                             3 3 3 exp 2 1 exp 2 1 exp 2 1 exp 2 1 exp 2 1 dkrrkj dkdkdkzzkyykxxkj dkzzjkdkyyjkdkxxjk zzyyxxrr SF zyxSFzSFySFx zSFzySFyxSFx SFSFSFSF       zyx zyx dkdkdkdk zkykxkk    3 111  where Paul Dirac 1902-1984 Joseph Fourier 1768-1830
  • 248. 248 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 1) In the same way: (continue – 4)            dttjtt 'exp 2 1 ' Therefore               'expexp 2 1 ' 3 4 ttjrrkjddkttrr SFSF     Let use the Fourier Transformation to write             'expexp,',;, 3 ttjrrkjkgddktrtrG SFFS   Hence                            'expexp 2 4 'expexp, 1 3 4 3 2 2 2 2 ttjrrkjddkttjrrkjkgddk tv SFSFS      or                                  'expexp 2 4 'expexp 1 exp'exp, 3 4 2 2 2 23 ttjrrkjddk ttj t rrkj v rrkjttjkgddk SF SFSFS      
  • 249. 249 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 2) Let compute: (continue – 5)                                         SFSFSFS SFzSFySFxSzyx SFzSFySFxzyxS SFzSFySFxSSSFS rrkjkrrkjkjkjrrkjkj zzkyykxxkjzjkyjkxjk zzkyykxxkjzjkyjkxjk zzkyykxxkjrrkj                           expexpexp exp111 exp111 expexp 2 2      'exp'exp 2 2 2 ttjttj t     Therefore:                            'expexp 2 4 'expexp, 3 4 2 2 23 ttjrrkjddk ttjrrkj v kkgddk SF SF         Because this is true for all k and ω, we obtain                 2 2 2 3 1 4 1 , v k kg   
  • 250. 250 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 3) (continue – 6)                                  'expexp 4 1 'expexp,',;, 2 2 2 3 3 3 ttjrrkj v k d dkttjrrkjkgddktrtrG SFSFFS       We can see that the integral in k has to singular points for v k   Let consider only the progressive wave, i.e. G = 0 for t > t’. To find the integral let change ω by ω + jδ where δ is a small negative number                           rkj v j k d dktrtrG FS      exp 4 1 ',;, 2 2 2 3 3 where and .SF rrr   'tt 
  • 251. 251 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 4) (continue – 7) In the plane ω we close the integration path by the semi-circle with and the singular points on the upper side, for τ > 0 (for t > t’)                           rkj v j k d dktrtrG FS      exp 4 1 ',;, 2 2 2 3 3 r      '00exp ttdjf UPC        '00exp ttdjf DOWNC                                 0exp 0exp exp      DOWN UP C C djf djf djf jvk  jvk  Re Im
  • 252. 252 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 5) (continue – 8)                                                               CC jvkjvk rkjv drkj v j k d rkj v j k d I           exp expexp 2 2 2 2 2 2 2 Let use the Cauchy Integral for a complex function f (z) continuous on a closed path C, in the complex z plane:      0 0 2lim2 0 zfjzfjdz zz zf zz C     We have:                       k kvrkj v vk jkv vk jkv rkjvj jvk rkjv j jvk rkjv jI vkvk            sinexp 2 2 exp 2 exp exp2 exp 2lim exp 2lim 2 2 0, 2 0,                      Therefore, we can write:                             k vkrkj dk v v k rkj ddktrtrG FS      sinexp 2 exp 4 1 ',;, 3 2 2 2 2 3 3  
  • 253. 253 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 6) (continue – 9) Let use spherical coordinates relative to vector:                             k vkrkj dk v v k rkj ddktrtrG FS      sinexp 2 exp 4 1 ',;, 3 2 2 2 2 3 3   r                  rr r r kk kk kk z y x z y x 0 0 cos sinsin cossin     dk sin dk dk     dddkkdk sin23    r x y z
  • 254. 254 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 7) (continue – 10)  kvd v r jkv v r jkv v r jkv v r jkv r                                                      4 expexpexpexp 1   r v rr tt v rr tt SFSF                  ''  r v r v r dx v r jx v r jx r                                                    expexp 2 11    kvd v r jkv v r jkv r v kvd v r jkv v r jkv r                                                          4 expexp 4 expexp 1                               dk j jvkjvk j jkrjkr r v 2 expexp 2 expexp               dkvkvkr r v dkvkvkr r v     sinsinsinsin 2 0                         00 0 sin 2 expexp2cosexp sin dkvk j jkrjkr r v dk jkr jkr vkk v                0 0 2 cossincosexp2 2    dkdvkjkrk v          0 0 2 0 2 2 sin sincosexp 2      dkddk k vkjkrv          k vkrkj dk v trtrG FS   sinexp 2 ',;, 3 2                   rr r r kk kk kk z y x z y x 0 0 cos sinsin cossin     dk sin dk dk     dddkkdk sin23    r x y z
  • 255. 255 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION • GREEN’s FUNCTION (continue – 8) (continue – 11) We can see that represents a progressive wave and represents a regressive wave:                    v rr tt v rr tt SFSF  ''                     v rr tt v rr tt SFSF  ''  Hence   SF SFSF FS rr v rr tt v rr tt trtrG                         '' ',;,  We shall consider only the progressive wave and use:   SF SF FS rr v rr tt trtrG                ' ',;,  Retarded Green Function The other solution is:   SF SF FS rr v rr tt trtrG                ' ',;,  Advanced Green Function
  • 256. 256 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 12) Gustav Robert Kirchhoff 1824-1887 Let return to the Helmholtz Non-homogeneous Differential Equation for the Electric Scalar Potential Ve is:      trtrV tv trV eee , 1 , 1 , 2 2 2 2        We want to find the Electric Scalar Potential Ve at the point F (field) due to all the sources (S) in the volume V, including its boundaries  n i iSS 1  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   F inside V F on the boundary of V Hermann von Helmholtz 1821-1894
  • 257. 257 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 13) Since is no defined at r = 0 we define the volume V’ as the volume V minus a small sphere of radius and surface around the point F, when F is inside V, or a semi-sphere of radius and surface around the point F, when F is on the boundary of V.  rG 00 r 00 r 2 04 rSF  2 02 rSF  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Let define the operator that acts only on the source coordinate .Sr            z z y y x x SSS S 111 Using the Green’s Identity                      FSS SFSSeSeSSF V SFSSeSeSSF dStrtrGtrVtrVtrtrGdVtrtrGtrVtrVtrtrG ',;,',',',;,',;,',',',;, ' 22  substitute here      ', 1 ', ' 1 ', 2 2 2 2 trtrV tv trV SeSeSeS               '4',;, ' 1 ',;, 2 2 2 2 ttrrtrtrG tv trtrG FSFSFSS       we obtain                                                   FSS SFSSeeSSSF V SFSe S SF V SFSeSeSF dStrtrGtrVtrVtrtrG dVttrrtrV tr trtrG dVtrtrG t trVtrV t trtrG v ',;,',',',;, '4', ', ',;, ',;, ' ',', ' ',;, 1 ' ' 2 2 2 2 2        
  • 258. 258 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 13) Let integrate the equation between to and choose t such that:1' tt  2' tt  21 /' tvrttt                              2 2 1 1 2 1 ''4', ', ',;, '',;, ' ',', ' ',;, 1 ' ' 2 2 2 2 2 I t t V SFSe S SF I t t V SFSeSeSF dtdVttrrtrV tr trtrG dtdVtrtrG t trVtrV t trtrG v                                                  4 2 1 3 2 1 '',;,',',',;, '',;,',',',;, I t t S SFSSeeSSSF I t t S SFSSeeSSSF dtdStrtrGtrVtrVtrtrG dtdStrtrGtrVtrVtrtrG F       
  • 259. 259 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 14) Integral I1     0 ' ',;,',;, 21            SF SF SFSF rr v rr tt ttrtrGttrtrG     Since 21 /' tvrttt  then                         0',;, ' ',', ' ',;, 1 '',;, ' ',', ' ',;, ' 1 '',;, ' ',', ' ',;, 1 ' 2 ' 2 ' 2 2 2 2 21 2 1 2 1 2 1                                           V t t SFSeSeSF V t t SFSeSeSF t t V SFSeSeSF dVtrtrG t trVtrV t trtrG v dVdttrtrG t trVtrV t trtrG tv dVdttrtrG t trVtrV t trtrG v I        0',;, ' ',;, ' 21       ttrtrG t ttrtrG t SFSF 
  • 260. 260 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 15) Integral I2                  2 1 ''4', ', ',;, ' 2 t t V SFSe S SF dtdVttrrtrV tr trtrGI                               2 1 2 1 '' ''',4' ',/' t t t t V SFSe V S FS dVdtttrrtrVdtdV tr rr vrtt                        ' /' 0 '' /' ',1 '',4 ',1 V vrttFS S V SFSe V vrttFS S dV rr tr dVttrrtrVdV rr tr              The integral is zero since in V’.     ' /', V FSSe dVrrvrttrV   FS rr   iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
  • 261. 261 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 16) Integral I3            2 1 '',;,',',',;,3 t t S SFSSeSeSSF dSdttrtrGtrVtrVtrtrGI                                              S t t SF SF SSeSeS SF SF dSdt rr v rr tt trVtrV rr v rr tt2 1 ' ' ',', '                                                  S t t SF S SF SeSeS SF SF dSdt rrv rr tttrVtrV rr v rr tt2 1 ' 1 '',', '                                      S t t SF SF S Se dSdt rr v rr tt trV 2 1 ' ' ',    
  • 262. 262 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 17) Integral I3 (continue – 1) But     v rr vrtt t rrvrtt rv rr tt SFS SFS rrr SF S SF                       /' ' /''  and                                                       S t t SF SF SFS Se S t t SF SF S Se dSdt v rr tt trrv rr trVdSdt rr v rr tt trV 2 1 2 1 '' ' ',' ' ',                                                   S t t SF SF SFS Se S t t SF SF SFS Se partsbyegrating dSdt v rr tt rrv rr trV t dS v rr tt rrv rr trV 2 1 2 1 ''', ' '', 0 int            
  • 263. 263 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 18) Integral I3 (continue – 2) Therefore                                             S t t SF Se SF SFS SF SSe SF SeS dSdt v rr tttrV trrv rr rr trV rr trV I 2 1 ''', ' 1 ', ', 3                                             S vrtt Se SF SFS SF SSe SF SeS dStrV trrv rr rr trV rr trV /' ', ' 1 ', ',                                                S vrtt Se SF SF SF SSe SF Se dStrV trrv rr n rr trV rr n trV /' ', ' 1 ', ',        The last equality follows from dS n U dSnn n U dSU SS             11
  • 264. 264 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 19) Integral I4 In the same way as for integral I4 we have             2 1 '',;,',',',;,4 t t S SFSSeSeSSF F dSdttrtrGtrVtrVtrtrGI                                     FS vrtt Se SF SFS SF SSe SF SeS dStrV trrv rr rr trV rr trV /' ', ' 1 ', ',                                                FS vrtt Se SF SF SF SSe SF Se dStrV trrv rr n rr trV rr n trV /' ', ' 1 ', ',       
  • 265. 265 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 20) Integral I4 (continue – 1) We use since points inside V normal to and points outside V.   n r r 1 0 0  0 0 r r  nS  n1 On the sphere or the semi-spherearound the field point F with radius and surface or if the point F is inside V or on the boundary, respectively, we have  FS rrr   00 2 04 rSF  2 02 rSF          n r r rr rr rr rr rr SF SF SF FS SSFS F 1 0 0                   n rr r rrr rr rrrr rr rrrr SF SF SFSF FS SFSSF S F 1 11111 2 00 0 2 0 22         ndr r r drdS FS 12 0 0 02 0  Since we can assume mean values for all field quantities in the integral00 r iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
  • 266. 266 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 21) Integral I4 (continue – 2)                                    FS vrtt Se SF SFS SF SSe SF SeS dStrV trrv rr rr trV rr trV I /' 4 ', ' 1 ', ',                              drnn rv trV t n r trV r trV vrtt SeSe SeS r 2 0 /'0 2 00 0 11 1 ', ' 1 1 ', ', lim 0                   0 0 /' 0/'0 0 0/'0 ', ' lim',lim1',lim 000                   d v r trV t dtrVdrntrV vrtt Se rvrttSe rvrttSeS r    trV SonF VinF trV SonFd VinFd FeFe , 2 4 ,2 0 4 0                                  iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr  
  • 267. 267 ELECTROMAGNETICSSOLO KIRCHHOFF’s SOLUTION OF THE SCALAR HELMHOLTZ NONHOMOGENEOUS DIFFERENTIAL EQUATION (continue – 22) SUMMARIZE The Kirchhoff’s solution to the Helmholtz Non-homogeneous Differential Equation:      trtrV tv trV eee , 1 , 1 , 2 2 2 2        is                                             S v rr tt Se SF SFS SF SSe SF SeS V v rr ttSF S Fe dStrV trrv rr rr trV rr trVT dV rr trT trV SFSF            '' ', ' 1 ', ', 4 ', 4 ,                                                         S v rr tt Se SF SF SF Se SF Se V v rr ttSF S dSndS n n dStrV trrv rr n rrn trV rr n trV T dV rr trT SF SF S            ' ' ˆ ˆ ', ' 1 ', ', 4 ', 4    Voutsiden SonF VinF T       1 2 1 1 iS nS  n i iSS 1 dV dSn  1 V Fr  Sr  F 0r SF rrr   iS nS dV dSn  1 V Fr  Sr  F 0r SF rrr   Table of Contents
  • 268. 268 SOLO References M.R. Spiegel, “Vector Analysis and an Introduction to Tensor Analysis”, Schaum’s Outline Series, McGraw-Hill, 1959 Vector Analysis H.Lass, “Vector and Tensor Analysis”, McGraw Hill, 1950 J,N, Reddy & M.L. Rasmussen, “Advanced Engineering Analysis”, John Willey, 1982, Ch.1:”Elements of Vector and Tensor Analysis” Table of Contents
  • 269. January 6, 2015 269 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA