![equilateral triangle =
triangle given SAS (two sides and the opposite angle)
= (1/2) a b sin C
triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's
formula)
regular polygon = (1/2) n sin(360°/n) S2
when n = # of sides and S = length from center to a corner
cube = a 3
rectangular prism = a b c
irregular prism = b h
cylinder = b h = pi r 2 h
pyramid = (1/3) b h
cone = (1/3) b h = 1/3 pi r 2 h](https://image.slidesharecdn.com/mathsformulae-121228112205-phpapp02/85/Maths-formulae-8-320.jpg)
![if w = r(COs + i sin );n=integer, then there are n complex nth
roots (z) of w for k=0,1,..n-1:
z(k) = r (1/n) [ COs( ( + 2(PI)k)/n ) + i sin( ( + 2(PI)k)/n ) ]
if z = r (COs + i sin ) then ln(z) = ln r + i
sin(a + BI) = sin(a)cosh(b) + COs(a)sinh(b) i
COs(a + BI) = COs(a)cosh(b) - sin(a)sinh(b) i
tan(a + BI) = ( tan(a) + i tanh(b) ) / ( 1 - i tan(a) tanh(b))
= ( sech 2(b)tan(a) + sec 2(a)tanh(b) i ) / (1 + tan 2(a)tanh 2(b))
FUNCTIONS
The fourier series of the function f(x)
a(0) / 2 + (k=1.. ) (a(k) cos kx + b(k) sin kx)
a(k) = 1/PI f(x) cos kx dx
b(k) = 1/PI f(x) sin kx dx
Remainder of fourier series. Sn(x) = sum of first n+1 terms
at x.
remainder(n) = f(x) - Sn(x) = 1/PI f(x+t) Dn(t) dt
Sn(x) = 1/PI f(x+t) Dn(t) dt
Dn(x) = Dirichlet kernel = 1/2 + cos x + cos 2x + .. + cos nx
= [ sin(n + 1/2)x ] / [ 2sin(x/2) ]
Riemann's Theorem. If f(x) is continuous except for a finite #
of finite jumps in every finite interval then:
lim(k-> ) f(t) cos kt dt = lim(k-> ) f(t) sin kt dt = 0](https://image.slidesharecdn.com/mathsformulae-121228112205-phpapp02/85/Maths-formulae-13-320.jpg)
![The fourier series of the function f(x) in an arbitrary
interval.
A(0) / 2 + (k=1.. ) [ A(k) cos (k(PI)x / m) + B(k) (sin
k(PI)x / m) ]
a(k) = 1/m f(x) cos (k(PI)x / m) dx
b(k) = 1/m f(x) sin (k(PI)x / m) dx
Parseval's Theorem. If f(x) is continuous; f(-PI) = f(PI) then
1/PI f^2(x) dx = a(0)^2 / 2 + (k=1.. ) (a(k)^2 +
b(k)^2)
Fourier Integral of the function f(x)
f(x) = ( a(y) cos yx + b(y) sin yx ) dy
a(y) = 1/PI f(t) cos ty dt
b(y) = 1/PI f(t) sin ty dt
f(x) = 1/PI dy f(t) cos (y(x-t)) dt
Special Cases of Fourier Integral
if f(x) = f(-x) then
f(x) = 2/PI cos xy dy f(t) cos yt dt
if f(-x) = -f(x) then
f(x) = 2/PI sin xy dy sin yt dt
Fourier Transforms
Fourier Cosine Transform](https://image.slidesharecdn.com/mathsformulae-121228112205-phpapp02/85/Maths-formulae-14-320.jpg)
Maths formulae
- 1. Closure Property of Addition Sum (or difference) of 2 real numbers equals a real number Additive Identity a+0=a Additive Inverse a + (-a) = 0 Associative of Addition (a + b) + c = a + (b + c) Commutative of Addition a+b=b+a Definition of Subtraction a - b = a + (-b) Closure Property of Multiplication Product (or quotient if denominator 0) of 2 reals equals a real number Multiplicative Identity a*1=a Multiplicative Inverse a * (1/a) = 1 (a 0) (Multiplication times 0) a*0=0 Associative of Multiplication
- 2. (a * b) * c = a * (b * c) Commutative of Multiplication a*b=b*a Distributive Law a(b + c) = ab + ac Definition of Division a / b = a(1/b) Conic Section The General Equation for a Conic Section: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 The type of section can be found from the sign of: B2 - 4AC If B2 - 4AC is... then the curve is a... <0 ellipse, circle, point or no curve. =0 parabola, 2 parallel lines, 1 line or no curve. >0 hyperbola or 2 intersecting lines. The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k). Circle Ellipse Parabola Hyperbola Equation (horiz. x2 / a2 + y2/ x2 + y2 = r2 4px = y2 x2 / a2 - y2 / b2= 1 vertex): b2 = 1 Equations of y = ± (b/a)x Asymptotes: Equation (vert. y2 / a2 + x2/ x2 + y2 = r2 4py = x2 y2 / a2 - x2 / b2= 1 vertex): b2 = 1
- 3. Equations of x = ± (b/a)y Asymptotes: a = major radius (= 1/2 length major a = 1/2 length axis) p = distance major axis b = minor r = circle from vertex to b = 1/2 length Variables: radius (= 1/2 radius focus (or minor axis length minor directrix) c = distance axis) center to focus c = distance center to focus Eccentricity: 0 c/a 1 c/a Relation to Focus: p=0 a2 - b2 = c2 p=p a2 + b2 = c2 Definition: is the sum of distance to difference distance to locus of all points distances to focus = between distances the origin is which meet the each focus is distance to to each foci is constant condition... constant directrix constant Geometry Related Topics: section on Circles (a+b) 2 = a 2 + 2ab + b 2
- 4. (a+b)(c+d) = ac + ad + bc + bd a 2 - b 2 = (a+b)(a-b) (Difference of squares) a3 b 3 = (a b)(a 2 ab + b 2) (Sum and Difference of Cubes) x 2 + (a+b)x + AB = (x + a)(x + b) if ax 2 + bx + c = 0 then x = ( -b (b 2 - 4ac) ) / 2a (Quadratic Formula) Powers x a x b = x (a + b) x a y a = (xy) a (x a) b = x (ab) x (a/b) = bth root of (x a) = ( bth (x) ) a x (-a) = 1 / x a x (a - b) = x a / x b Logarithms y = logb(x) if and only if x=b y logb(1) = 0 logb(b) = 1 logb(x*y) = logb(x) + logb(y) logb(x/y) = logb(x) - logb(y) logb(x n) = n logb(x)
- 5. logb(x) = logb(c) * logc(x) = logc(x) / logc(b) The Functions Trig Functions: sine(q) = opp/hyp cosecant(q) = hyp/opp cosine(q) = adj/hyp secant(q) = hyp/adj tangent(q) = opp/adj cotangent(q) = adj/opp The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot). It is often simpler to memorize the the trig functions in terms of only sine and cosine: sine(q) = opp/hyp csc(q) = 1/sin(q) cos(q) = adj/hyp sec(q) = 1/COs(q) tan(q) = sin(q)/cos(q) cot(q) = 1/tan(q) Inverse Functions arcsine(opp/hyp) = q arccosecant(hyp/opp) = q arccosine(adj/hyp) = q arcsecant(hyp/adj) = q arctangent(opp/adj) = q arccotangent(adj/opp) = q ‘ Sine: Integral sin(x) dx = -cos(x) + C Sine: Derivative (sin(x)) = cos(x)
- 6. Sine: Expansions Series: sin(x) = (-1)k x2k+1 / (2k+1)! = x - (1/3!)x3 + (1/5!)x5 - (1/7!)x7 (This can be derived from Taylor's Theorem.) Product: sin(x) = x (1 - (x / kPI)2) = x(1 - (x/PI)2)(1 - (x/2PI)2)(1 - (x/3PI)2)*... Cosine: Expansions Series: cos(x) = (-1)k x2k / (2k)! = 1 - (1/2!)x2 + (1/4!)x4 - (1/6!)x6 (This can be derived from Taylor's Theorem.) Cosine: Derivative (cos(x)) = -sin(x) Cosine: Integral cos(x) dx = -sin(x) + C
- 7. (pi = = 3.141592...) Area Formulas Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a". Be careful!! Units count. Use the same units for all measurements. Examples square = a 2 rectangle = ab parallelogram = bh trapezoid = h/2 (b1 + b2) circle = pi r 2 ellipse = pi r1 r2 one half times the base length times the triangle = height of the triangle
- 8. equilateral triangle = triangle given SAS (two sides and the opposite angle) = (1/2) a b sin C triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula) regular polygon = (1/2) n sin(360°/n) S2 when n = # of sides and S = length from center to a corner cube = a 3 rectangular prism = a b c irregular prism = b h cylinder = b h = pi r 2 h pyramid = (1/3) b h cone = (1/3) b h = 1/3 pi r 2 h
- 9. sphere = (4/3) pi r 3 ellipsoid = (4/3) pi r1 r2 r3 Diameter = 2 x radius of circle Circumference of Circle = PI x diameter = 2 PI x radius where PI = = 3.141592... Area of Circle: area = PI r2 Length of a Circular Arc: (with central angle ) if the angle is in degrees, then length = x (PI/180) x r if the angle is in radians, then length = r x Area of Circle Sector: (with central angle ) if the angle is in degrees, then area = ( /360)x PI r2 if the angle is in radians, then area = (( /(2PI))x PI r2 Equation of Circle: (Cartesian coordinates) for a circle with center (j, k) and radius (r): (x-j)^2 + (y-k)^2 = r^2 Equation of Circle: (polar coordinates) for a circle with center (0, 0): r( ) = radius
- 10. for a circle with center with polar coordinates: (c, ) and radius a: r2 - 2cr cos( - ) + c2 = a2 Equation of a Circle: (parametric coordinates) for a circle with origin (j, k) and radius r: x(t) = r cos(t) + j y(t) = r sin(t) + k VECTORS |<a, b>| = magnitude of vector = (a 2+ b 2) |<x1, .., xn>| = (x12+ .. + xn2) <a, b> + <c, d> = <a+c, b+d> <x1, .., xn> + <y1, .., yn>= < x1+y1, .., xn+yn> k <a, b> = <ka, kb>
- 11. k <x1, .., xn> = <k x1, .., k x2> <a, b> <c, d> = ac + bd <x1, .., xn> <y1, ..,yn> = x1 y1 + .. + xn yn> R S= |R| |S| cos ( = angle between them) R S= S R (a R) (bS) = (ab) R S R (S + T)= R S+ R T R R = |R| 2 |R x S| = |R| |S| sin ( = angle between both vectors). Direction of R xS is perpendicular to A & B and according to the right hand rule. | i j k | R x S = | r1 r2 r3 | = / |r2 r3| |r3 r1| |r1 r2| | s1 s2 s3 | |s2 s3| , |s3 s1| , |s1 s2| / SxR=-RxS (a R) x S = R x (a S) = a (Rx S) R x (S + T) = R x S + Rx T RxR=0 If a, b, c = angles between the unit vectors i, j,k and R Then the direction cosines are set by: COs a = (R i) / |R|; COs b = (R j) / |R|; COs c = (R k) / |R|
- 12. |R x S| = Area of parrallagram with sides Rand S. Component of R in the direction of S = |R|COs = (R S) / |S| (scalar result) Projection of R in the direction of S = |R|COs = (R S) S/ | S| 2 (vector result) Basic Operations i= (-1) i 2 = -1 1 / i = -i i 4k = 1; i (4k+1) = i; i (4k+2) = -1; i (4k+3) = -i (k = integer) (i)= (1/2)+ (1/2) i Complex Definitions of Functions and Operations (a + bi) + (c + di) = (a+c) + (b + d) i (a + BI) (c + DI) = ac + adi + bci + bdi 2 = (ac - bd) + (ad +bc) i 1/(a + BI) = a/(a 2 + b 2) - b/(a 2 + b 2) i (a + BI) / (c + DI) = (ac + BD)/(c 2 + d 2) + (BC - ad)/(c 2 +d 2) i a2 + b2 = (a + BI) (a - BI) (sum of squares) e (i ) = cos + i sin n (a + BI) = (COs(b ln n) + i sin(b ln n))n a if z = r(COs + i sin ) then z n = r n ( COs n + i sin n ) (DeMoivre's Theorem)
- 13. if w = r(COs + i sin );n=integer, then there are n complex nth roots (z) of w for k=0,1,..n-1: z(k) = r (1/n) [ COs( ( + 2(PI)k)/n ) + i sin( ( + 2(PI)k)/n ) ] if z = r (COs + i sin ) then ln(z) = ln r + i sin(a + BI) = sin(a)cosh(b) + COs(a)sinh(b) i COs(a + BI) = COs(a)cosh(b) - sin(a)sinh(b) i tan(a + BI) = ( tan(a) + i tanh(b) ) / ( 1 - i tan(a) tanh(b)) = ( sech 2(b)tan(a) + sec 2(a)tanh(b) i ) / (1 + tan 2(a)tanh 2(b)) FUNCTIONS The fourier series of the function f(x) a(0) / 2 + (k=1.. ) (a(k) cos kx + b(k) sin kx) a(k) = 1/PI f(x) cos kx dx b(k) = 1/PI f(x) sin kx dx Remainder of fourier series. Sn(x) = sum of first n+1 terms at x. remainder(n) = f(x) - Sn(x) = 1/PI f(x+t) Dn(t) dt Sn(x) = 1/PI f(x+t) Dn(t) dt Dn(x) = Dirichlet kernel = 1/2 + cos x + cos 2x + .. + cos nx = [ sin(n + 1/2)x ] / [ 2sin(x/2) ] Riemann's Theorem. If f(x) is continuous except for a finite # of finite jumps in every finite interval then: lim(k-> ) f(t) cos kt dt = lim(k-> ) f(t) sin kt dt = 0
- 14. The fourier series of the function f(x) in an arbitrary interval. A(0) / 2 + (k=1.. ) [ A(k) cos (k(PI)x / m) + B(k) (sin k(PI)x / m) ] a(k) = 1/m f(x) cos (k(PI)x / m) dx b(k) = 1/m f(x) sin (k(PI)x / m) dx Parseval's Theorem. If f(x) is continuous; f(-PI) = f(PI) then 1/PI f^2(x) dx = a(0)^2 / 2 + (k=1.. ) (a(k)^2 + b(k)^2) Fourier Integral of the function f(x) f(x) = ( a(y) cos yx + b(y) sin yx ) dy a(y) = 1/PI f(t) cos ty dt b(y) = 1/PI f(t) sin ty dt f(x) = 1/PI dy f(t) cos (y(x-t)) dt Special Cases of Fourier Integral if f(x) = f(-x) then f(x) = 2/PI cos xy dy f(t) cos yt dt if f(-x) = -f(x) then f(x) = 2/PI sin xy dy sin yt dt Fourier Transforms Fourier Cosine Transform
- 15. g(x) = (2/PI) f(t) cos xt dt Fourier Sine Transform g(x) = (2/PI) f(t) sin xt dt Identities of the Transforms If f(-x) = f(x) then Fourier Cosine Transform ( Fourier Cosine Transform (f(x)) ) = f(x) If f(-x) = -f(x) then Fourier Sine Transform (Fourier Sine Transform (f(x)) ) = f(x)