Summary (chapter 1 chapter6)

Chapter 12 Vectors and the Geometry of Space
12.1 Three-dimensional Coordinate systems
A. Three dimensional Rectangular Coordinate Sydstem:

The Cartesian product
R3 = R × R × R = {(x, y, z) : x, y, z ∈ R},
where (x, y, z) iscalled ordered triple.

B. Distance:

The distance |P1 P2 | between two points P1 = (x1 , y1 , z1 ) and P2 = (x2 , y2 , z2 ) is
|P1 P2 | = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .

C. Sphere:

An equation of a sphere with center C(h, k, ) and radius r is
(x − h)2 + (y − k)2 + (z − )2 = r2 a.

12.2 Vectors
A. Vector: a quantity that has both of magnitude and direction.
−→
B. (Re)presentation and Notation: − , < a1 , a2 , a3 >, AB.
→
a

Deﬁnition 1. A two dimensional vector is an ordered pair − =< a1 , a2 > with a1 and
→
a
a2 real numbers. A three dimensional vector is an ordered triple − =< a1 , a2 , a3 >
→
a
with a1 , a2 and a3 real numbers.

C. Magnitude: Let − =< a1 , a2 , a3 >. Then, the magnitude of − is
→
a →
a

|− | =
→
a a2 + a2 + a2 .
1 2 3

D. Multiplication of a vector by a scalar:

Let − =< a1 , a2 , a3 > and c be a scalar (real number). Then,
→
a
c− =< ca1 , ca2 , ca3 > and |c− | = |c||− |.
→a →
a →
a
→
−
E. Vector addition/diﬀerence : Let − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >. Then,
→a
→ →
− + − =< a + b , a + b , a + b >,
a b 1 1 2 2 3 3

→ →
− − − =< a − b , a − b , a − b >
a b 1 1 2 2 3 3

F. Standard Basis vectors :

In R2 ,
→
− →
−
i =< 1, 0 >, j =< 0, 1 > .
In R3 ,
→
− →
− →
−
i =< 1, 0, 0 >, j =< 0, 1, 0 >, k =< 0, 0, 1 > .
→
− →
− →
−
If − =< a1 , a2 , a3 >, then − = a1 i + a2 j + a3 k .
→
a →
a

G. Unit Vector :

A vector whose length is 1. If − = 0, then the unit vector that has the same direction as
→
a
→
− is
a
→
−
− = a .
→u
|− |
→
a

12.3 The Dot Product
→
− →
−
Deﬁnition 2. If − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >, the dot product of − and b
→a →
a
→ −→
is the number − · b given by
a
→ →
− ·− =a b +a b +a b .
a b 1 1 2 2 3 3

Note 1: A result of the dot product of two vectors is a scalar (not a vector)
Note 2: The dot product is sometimes called inner product or scalar product.

A. Properties of the dot product

(1) → →
− · − = |− |2
a a →a
→ −
→ − − → →
− · b = b · a
(2) a
(3) → → →
− · (− + − ) = − · − + − · −
a b c → → → →
a b a c
(4) → →
− ) · − = c(− · − ) = − · (c− )
(c a b → →
a b →
a
→
b
→ →
− −
(5) 0 · a =0

→
−
Theorem 1. Let − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >. If θ is the angle between the
→a
→
−
vectors − and b , then
→
a
→ →
− · − = |− ||− | cos θ.
a b → →
a b
2

→
−
Corollary 1. Let − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >. If θ is the angle between the
→
a
→
−
vectors − and b , then
→
a
→ →
− ·−
a b
cos θ = .
→ →
− ||− |
|a b

B. Direction Angles and Direction Cosine:

Let the angles betwees − and x-axis, y-axis, and z-axis are α, β, and γ respectively. Then,
→
a
→
− =< a , a , a >,
if a 1 2 3
a1 a2 a3
cos α = − , cos β = −
→| →| cos γ = |− | .
→
|a |a a
Then,
a
− | =< cos α, cos β, cos γ > .
→
|a

C. Projection :

→
−
C.1. Scalar Projection : component of b along − →
a
→
− →
− → →
− ·−
a b
comp− b = | b | cos θ = − .
→
a
|→|
a

C.2. Vector Projection:
→
− →→
− −a → →
− ·−
a b →
−
a
proj− b = comp− b − =
→ → →| − |.
a a
|a |− |
→
a →
|a

12.4 The Cross Product
→
−
Deﬁnition 3. If − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >, the cross product of − and
→a →
a
→
− → →
− × − given by
b is the vector a b
→ →
− × − =< a b − a b , a b − a b , a b − a b > .
a b 2 3 3 2 3 1 1 3 1 2 2 1

→ −
a
→
Theorem 2. The cross product − × b is orthogonal to both of − =< a1 , a2 , a3 > and
→
a
→
−
b =< b1 , b2 , b3 >.

Note 1: A result of the cross product of two vectors is a vector (Not a scalar). So, the
cross product is sometimes called vector product.

3

→
−
Theorem 3. If θ, 0 ≤ θ ≤ π, is the angle between − =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >,
→
a
then
→ −
a
→ → →
a
−
|− × b | = |− || b | sin θ.

→ −→
Note 2: the length of the cross product − × b is equal to the area of the parallelogram
a
→
−
determined by − and b .
→
a

A. Scalar Triple Product :

→ −
a
→
The volume of the parallelopiped determined by the vectors − , b , and − is the magnitude
→
c
of the scalar triple product
→ − →→
|− · ( b × − )|.
a c

12.5 Equations of Lines and Planes
A. Equation of a line L :

A line in a space is determined by a point P0 (x0 , y0 , z0 ) a vector − that is parallel to the
→
n
line. Let − =< a, b, c >, − =< x, y, z >, − 0 =< x0 , y0 , z0 >, and t is a scalar.
→
v →
r →
r

A.1. Vector Equation : − = − 0 + t− .
→ →
r r →
v
A.2. Parametric Equation : x = x0 + at y = y0 + bt z = z0 + ct.
A.3. Symmetric Equation :
x − x0 y − y0 z − z0
= = .
a b c

B. Equation of a Plane :

A plane in a space is determined by a point P0 (x0 , y0 , z0 ) a vector − (normal vector) that
→
n
is orthogonal to the plane. Let − =< x, y, z >, − 0 =< x0 , y0 , z0 >, and − =< a, b, c >.
→r →r →n

B.1. Vector Equation of a Plane:
→ → →
− · (− − − ) = 0.
n r r0

B.2.. Scalar Equation of a plane :
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
(or ax + by + cz + d = 0 with d = −ax0 − by0 − cz0 ).
4

C. Distance D from a point P1 (x1 , y1 , z1 ) to the plance ax + by + cz + d = 0:
|ax1 + by1 + cz1 + d|
D= √ .
a2 + b 2 + c 2

12.6/7 Cylinder and Cylindrical Coordinates
Identify and sketch the surfaces
(1) x2 + y 2 = 1 (2) y 2 + z 2 = 1.

To convert from cylindrical to rectangular Coordinate,
x = r cos θ y = r sin θ z = z.
To convert from rectangular to cylindrical Coordinate,
y
r2 = x2 + y 2 tan θ = z = z.
x

Chapter 13 Vector Functions
13.1 Vector Functions and Space Curves
A. Vector Function: A vector (valued) function (e.g, in R3 ) is of the form
→
− (t) =< f (t), g(t), h(t) >,
r
where all the component functions f , g, and h are real valued function.

B. Limit of a vector function − : If − (t) =< f (t), g(t), h(t) >, then
→
r →
r
→
− (t) =< lim f (t), lim g(t), lim h(t) >
lim r
t→a t→a t→a t→a

provided the limits of f (t) ,g(t), and h(t) exist.

In particular, a vector function − is continuous at t = a
→
r
lim − (t) = − (a).
→
r →
r
t→a

C. Space Curve: Suppose that f , g, and h are continuous functions on an interval I. Let
C = {(x, y, z) : x = f (t) y = g(t) z = h(t)},
where t varies through the interval I, is called a space curve.

The equations x = f (t), y = g(t), z = h(t) are called parametric equations of C and t
is called a parameter.
5

13.2 Derivatives and Integrals of Vector Functions
A. Derivative : The derivative of a vector (valued) function − is defined by
→
r

d−
→r →
− (t + h) − − (t)
r →
r
= − (t) = lim
→
r
dt h→0 h
if the limit exists.

The vector − (t) is called the tangent vector of − , and it unit tangent vector is given
→
r →r
by
→
− (t)
r
T(t) = − → (t)| .
|r
Theorem 4. If r→
− (t) =< f (t), g(t), h(t) >, where f , g, and h are differentiable functions,
then
→
− (t) =< f (t), g (t), h (t) > .
r

Theorem 5. Suppose − and − are differentiable vector functions, c is a scalar, and f is
→
u →
v
a real valued function. Then,

d −
(1) [→(t) + − (t)] = − (t) + − (t)
u →v →u →
v
dt
d −
(2) [c→(t)] = c− (t)
u →u
dt
d
(3) [f (t)− (t)] = f (t)− (t) + f (t)− (t)
→
u →u →
u
dt
d −
(4) [→(t) · − (t)] = − (t) · − (t) + − (t) · − (t)
u →v →
u →v →u →
v
dt
d −
(5) [→(t) × − (t)] = − (t) × − (t) + − (t) × − (t)
u →v →u →
v →
u →
v
dt
d −
(6) [→(f (t))] = f (t)− (f (t)) (Chain Rule)
u →u
dt

• See Example 1-5

B. Integrals : If − (t) =< f (t), g(t), h(t) >, where f , g, and h are integrable in [a, b], then
→r
the definite integral if the vector function − (t) can be defined by
→
r
b b b b
→
− (t)dt = →
− →
− →
−
r f (t)dt i + g(t)dt j + h(t)dt k.
a a a a

• See Example 6
6

13.3 Arc Length and Curvature
A. Arc Length : Let a ≤ t ≤ b, and let − (t) =< f (t), g(t), h(t) > where f , g , and h
→
r
are continuous on I. Then, the length of the space curve (arc length) from t = a to t = b
is deﬁned by
b
L = →
− (t) 2 dt
r
a
b
= [f (t)]2 + [g (t)]2 + [h (t)]2 dt
a
b 2 2 2
dx dy dz
= + + dt
a dt dt dt

• See Example 1.

13.4 Motion in Space :Velocity and Acceleration
A. Velocity vector : Suppose a particle moves through space so that its position vector
at time t is − (t). The the velocity vector at time t is deﬁned by
→
r
→
− →
−
− (t) = lim r (t + h) − r (t) = − (t).
→v →r
h→0 h

The velocity vector − (t) is also the tangent vector and points in the direction of the tangent
→v
line. Further, the speed of the particle at time t is
→
− (t) = − (t) .
v →r

B. Acceleration vector : The acceleration of the particle at time t is
→
− (t) = − (t) = − (t).
a →
v →r

• See Examples 1-3.
C. Newton’s Second Law of Motion : If, at any time t, a force F(t) acts on an object
of mass m producing an acceleration − (t), then
→
a
F(t) = m− (t).
→
a

• See Examples 4 and 5.

7

Chapter 14 Partical derivatives

14.1 Functions of Several Variables
A. Functions of two varialbes:

Deﬁnition 4. A function f of two variables is a rule of the form
f : (x, y) ∈ D → z = f (x, y).
Here the set D is the domain of f and its range is the set {f (x, y) : (x, y) ∈ D}.

• See Examples 1, 4
B. Graph : Let f is a function of two variables with domain D. Then, the graph of f is
{(x, y, z) ∈ R3 : z = f (x, y), (x, y) ∈ D}.

14.2 Limits and Continuity
• Let’s think about the two limits

x2 − y 2 x2 − y 2
lim lim and lim lim .
x→0 y→0 x2 + y 2 y→0 x→0 x2 + y 2

A. Limit :

Deﬁnition 5. Let f be a function of two variables with domain D and (a, b) ∈ closure(D).
We say that the limit of f (x, y) as (x, y) → (a, b) is L, i.e.,
lim f : (x, y) = L
(x,y)→(a,b)

if for any number > 0 there is number δ > 0 such that f (x, y) − L < whenever
(x, y) − (a, b) < δ and (x, y) ∈ D.

Remark: Let f (x, y) → L1 as (x, y) → (a, b) along a path C1 and f (x, y) → L2 as
(x, y) → (a, b) along a path C2 . Thn, if L1 = L2 , the limit lim(x,y)→(a,b) f : (x, y) does not
exist.

8

B. Continuity :

Definition 6. Let f be a function of two variables with domain D and (a, b) ∈ closure(D).
The function f (x, y) is called continuous at (a, b) if
lim f : (x, y) = f (a, b).
(x,y)→(a,b)

We say f (x, y) is continuous on D if f is continuous at every points (a, b) in D.

14.3 Partial Derivatives
A. Definition : Let f be a function of two variables. Its partial derivatives are functions
fx and fy defined by
f (x + h, y) − f (x, y)
fx (x, y) = lim
h→0 h
f (x, y + h) − f (x, y)
fy (x, y) = lim
h→0 h

B. Notations: If z = f (x, y),
∂f ∂ ∂z
fx (x, y) = fx = = f (x, y) = = Dx f
∂x ∂x ∂x
∂f ∂ ∂z
fy (x, y) = fy = = f (x, y) = = Dz f
∂y ∂y ∂x

Theorem 6. Let f be a function of two variables and defined on D. Let (a, b) ∈ D. Then,
if fxy and fyx are both continuous on D,
fxy (a, b) = fyx (a, b).

14.4 Tangent Planes and Linear Approximation
A. Definition : Let f be a function of two variables, and assume that fx and fy are
continuous. An equation of tangent plane to the surface z = f (x, y) at P (x0 , y0 , z0 ) is
z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ).

Define L(x, y) = f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ). Then L(x, y) is called a
linearization of f at (x0 , y0 ). Also, the approximation
f (x, y) ≈ L(x, y)
is called the linear approximation and the thangent plane approximation.

9

Definition 7. Let z = f (x, y) and ∆z = f (x0 + ∆x, y0 + ∆y). Then the function f is
differentiable at (x0 , y0 ) if ∆z can be written
∆z = fx (x0 , y0 )∆x + fy (x0 , y0 )∆y + 1 δx + 2 δxy,

where 1 , 2 → 0 as δx, δy → 0.
Theorem 7. Asume that fx and fy exist near (x0 , y0 ) and are continuous at (x0 , y0 ). Then
f is differentiable at (x0 , y0 ).

Total differential: The differential or total differential is is defined by
∂z ∂z
dz = fx (x, y)dx + fy (x, y) = dx + dy.
∂x ∂y

• See Examples 1,2, 4

14.5 The Chain Rule
The Chain Rule (Case 1): Assume that z = f (x, y) is a differentiable function of x and
y, where x = g(t) and y = h(t) are both differentiable of t. Then
dz ∂f dx ∂f dy
= + .
dt ∂x dt ∂y dt

The Chain Rule (Case 2): Assume that z = f (x, y) is a differentiable function of x and
y, where x = g(s, t) and y = h(s, t) are both differentiable functions of s and t. Then
∂z ∂z dx ∂f dy
= +
∂s ∂x ds dy ds
∂z ∂z dx ∂f dy
= + .
∂t ∂x dt dy dt

• See Examples 1,3, 5( For General Version)
Implicit Differentiation : Assume that z = f (x, y) is given implicitly as a function of
∂F
the form F (x, y, z) = 0. If F and f are differentiable and = 0, then
∂z
∂F ∂F
∂z ∂z ∂y
= − ∂x =−
∂x ∂F ∂y ∂F
∂z ∂z
• See Examples 9

10

14.6 Directional Derivatives and the Gradient Vector
A. Definition : Let f be a function of two variables. The directional derivatives of
f (x0 , y0 ) in the direction of unit vector − =< a, b > is
→
u
f (x0 + ha, y + hb) − f (x0 , y0 )
D− f (x0 , y0 ) = lim
→
u
h→0 h
if the limit exists.
In the case of three variable function, we can define the directional derivatives in a similar
manner.
Theorem 8. Let f be a differentialbe function of x and y. Then f has directional deriva-
tives in the direction of unit vector − =< a, b > and
→
u
D− f (x, y) = fx (x, y)a + fy (x, y)b.
→
u

The Gradient Vector : Let f be a function of several (say three) variables. The Gradient
of f is the vector function f fdefined by
→ ∂f − ∂f −
∂f − → →
f (x, y, z) =< fx (x, y, z), fy (x, y, z), fz (x, y, z) >= i + j k
∂x ∂x ∂z

Note that for any − =< a, b, c >, D− f (x, y, z) =
→
u →
u f (x, y, z) · − .
→
u

• See Examples 2, 3, 4

14.7 Maximum and Minimum Values

Definition 8. Let f be a function of two variables. Then, f (a, b) is called local maximum
value if f (a, b) ≥ f (x, y) when (x, y) is near (a, b). Also, f (a, b) is called local minimum
value if f (a, b) ≤ f (x, y) when (x, y) is near (a, b).

Theorem 9. If f has local maximum or minimum value at (a, b) and fx and fy exist, then
fx (a, b) = 0 and fy (a, b) = 0.

A point (a, b) is called a critical point of f if fx )a, b) = 0 and fy (a, b) = 0.

Second Derivatives Test: Assume that the second partial derivatives of f are continuous
on a disk with center (a, b), and assume that fx (a, b) = 0 and fy (a, b) = 0. Let
D = fxx (a, b)fyy (a, b) − [fxy (a, b)]2 .
(a) If D > 0 and fxx (a, b) > 0, then f (a, b) is a local minimum.

(b) If D > 0 and fxx (a, b) < 0, then f (a, b) is a local maximum.
11

(c) If D < 0, then f (a, b) is not a local maximum or minimum. (the point (a,b) is called a
saddle point of f ).
• See Examples 1, 2, 3, 6.

Theorem 10. If f is continuous on a closed, bounded set D in R2 , then f attains an
absolute maximum value f (x1 , y1 ) and an absolute minimum value f (x2 , y2 ) at some points
(x1 , y1 ) and (x2 , y2 ) in D.

To find an absolute maximum and minimum values of f on a closed, bounded set D :
(1) Find the values of f at the critical points of f in D.
(2) Find the extreme values of f on the boundary of D.
(3) The largest of the values from steps 1 and 2 is the absolute maximum value; The
smallest of the values is the absolute minimum value.
• See Example 7.

14.8 Lagrange Multipliers
Method of Lagrange Multifiers : In order to find the maximum and minimum values
of f (x, y, z) subject to the constraint g(a, y, z) = k
(a) Find all values of x, y, z, and λ such that
f (x, y, z) = λ g(x, y, z)
g(a, y, z) = k

(b) Evaluate f at all the points (x, y, z) that results from step (a). The largest of the values
is the absolute maximum value; The smallest is the absolute minimum value of f .

12

Chapter 15 Multiple Integrals
15.1 Double Integrals over Rectangles
Deﬁnition 9. The double integral of f over R = [a, b] × [c, d] is
∞ ∞
f (x, y)dA = lim f (xi , yj )∆A (1)
R m,n→∞
i=1 j=1

if the limit exists, where (xi , yj ) is in
Rij = [xi−1 , xi ] × [yj−1 , yj ].
Here the right-hand side of (1) is called s double Riemann sum.

If f (x, y) ≥ 0, the volume V of the solid that lies above the rectangle R and below the
surface z = f (x, y) is
V = f (x, y)dA.
R

15.2 Iterated Integrals
Theorem 11. (Fubini) Let f be a continuous function on R = [a, b] × [c, d].
b d d b
f (x, y)dA = f (x, y)dydx = f (x, y)dxdy.
R a c c a
The two integrals in the right-hand side of the above identity are called iterated integrals.
More generally, this theorem is true if f is bounded on R, f is discontinuous only on a
ﬁnite number if snmooth curves, and the iterated integrals exist.

Special Cases : If f (x, y) = g(x)h(y) on R = [a, b] × [c, d],
b d d b
f (x, y)dA = f (x, y)dydx = h(y)dy · g(x)dx .
R a c c a


15.3 Double Integrals over General Regions
Type I : Let f be a continuous on a type I region D such that
D = {(x, y)|a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x)}
then
b g2 (x)
f (x, y)dA = f (x, y)dydx.
R a g1 (x)
13

Type II : Let f be a continuous on a type II region D such that
D = {(x, y)|c ≤ y ≤ d, h1 (x) ≤ x ≤ h2 (x)}
then
d h2 (x)
f (x, y)dA = f (x, y)dxdy.
R c h1 (x)


Properties of Double Integrals : Assume that all of the following integrals exist. Then,

(1) [f (x, y) + g(x, y)]dA = f (x, y)dA + [g(x, y)]dA
D D D

(2) cf (x, y)dA = c [f (x, y)]dA
D D

(3) f (x, y)dA ≥ [g(x, y)]dA, if f (x, y) ≥ g(x, y)
D D

(4) 1dA = A(D)
D

(4) f (x, y)dA ≥ [g(x, y)]dA, + [g(x, y)]dA, if D = D1 ∪ D2 .
D D1 D2

Here D1 and D2 don’t overlap except (perhaps) on the boundary. Also, if m ≤ f (x, y) ≤ M
for all (x, y) ∈ D, then
mA(D) f (x, y)dA ≤ M A(D).
D

15.4 Double Integrals over Polar Coordinates
Change to Polar Coordinates in a Double Integral : Let f be a continuous on a po-
lar rectangle
R = {(r, θ) | 0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β}
then
β b
f (x, y)dA = f (r cos θ, r sin θ)rdrdθ).
R α a

• See Examples 1,2.

15.7 Triple Integrals
14

Theorem 12. (Fubini’s Theorem for Triple Integrals) Let f be a continuous function on
B = [a, b] × [c, d] × [r, s].
s d b d b
f (x, y, z)dV = f (x, y, z)dxdydz = f (x, y)dxdy.
B r c a c a

Triple Integrals over General Regions:

Type I : Let f be a continuous on region D (type I or type II in double integral) such
that
E = {(x, y, z) | (x, y) ∈ D u1 (x, y) ≤ z ≤ u2 (x, y)},
then
u2 (x,y)
f (x, y, z)dV = f (x, y, z)dz dA.
E D u1 (x,y)

Type II : Let f be a continuous on region D (type I or type II in double integral) such
that
E = {(x, y, z) | (y, z) ∈ D u1 (y, z) ≤ z ≤ u2 (y, z)},
then
u2 (y,z)
f (x, y, z)dV = f (x, y, z)dx dA.
E D u1 (y,z)


15.8 Triple Integrals in Cylindrical and Spherical Coordinates

Formula for triple integration in cylindrical coordinates:
u2 (r cos θ,r sin θ)
f (x, y, z)dV = f (r cos θ, r sin θ, z)rdzdrdθ.
E D u1 (r cos θ,r sin θ)

• See Examples 1,2.

15

Chapter 16 Vector Calculus
16.1 Vector Fields
Definition 10. Let E be a subset of R3 . A vector field on R3 is a function F that assigns
to each (x, y, z) ∈ E a three-dimensional vector F(x, y, z). We can write F as follows:
→
− →
− →
−
F(x, y, z) = P (x, y, z) i + Q(x, y, z) j + R(x, y, z) k .

Gradient Fields: If f is a scalar function of three (or two) variables, its gradient is a
vector field on R3 given by
→
− →
− →
−
f (x, y, z) = fx (x, y, z) i + fy (x, y, z) j + fz (x, y, z) k .
• See Examples 1, 2, 6.

16.2 Line Integrals
Definition 11. Let C be a smooth curve given by the parametric equation
x = x(t) y = y(t), a ≤ t ≤ b.
If f is defined on the curve C, then the line integral of f along C is defined by
b 2 2
dx dy
f (x, y)ds = f (x(t), y(t)) + dt.
C a dt dt

Remark: If C is a piecewise-smooth curve, that is, C is a finite union of smooth
curves C1 , · · · Cn , then

f (x, y)ds = f (x, y)ds + · · · + f (x, y)ds.
C C1 Cn

Line integral of f along C with respect to x and y:
b
f (x, y)dx = f (x(t), y(t))x (t)dt
C a
b
f (x, y)dy = f (x(t), y(t))y (t)dt
C a

• See Examples 1, 2, 4.

Line integrals in Space: Suppose that C is a smooth curve given by the parametric
equation
x = x(t) y = y(t) z = z(t), a ≤ t ≤ b.
16

If f is defined on the curve C, then the line integral of f along C is defined by
b 2 2 2
dx dy dz
f (x, y)ds = f (x(t), y(t), z(t)) + + dt.
C a dt dt dt

Compact Notation :
b
f (x, y, z)ds = f (− (t))|− (t)|dt.
→
r →
r
C a

• See Examples 5, 6.

Line integrals of Vector Fields: Let F be a continuous vector field defined on a smooth
curve C given by a vector function − (t), a ≤ t ≤ b. Then, the line integral of F along
→
r
C is
b
F · d− =
→
r F(− (t)) · − (t)dt =
→r →
r F · T ds,
C a C
where T (x, y, z) is the unit tangent vector at the point (x, y, z).

• See Examples 7, 8.

16.3 The Fundamental Theorem for Line Integrals

Theorem 13. Let C be a smooth curve given by the vector function − (t), a ≤ t ≤ b. Let
→
r
f be a continuous function and its f is continuous on C. Then,

f · d− = f (− (b)) − f (− (a)).
→
r →
r →
r
C

Note: We can evaluate f · d− by knowing the value of f at the end of points of C.
→
r
C

Definition 12. A vector field F is called a conservative vector field if there is a scalar
function f such that F = f . Here f is called a potential function of F.

Note: Line integrals of conservative vector fields are independent of path.

Theorem 14. f · d− is independent of path in a domain D iff
→
r f · d− = 0 for
→
r
C C
every closed path in D.

17

→
− →
−
Theorem 15. Let F = P i + Q j be a conservative vector field, where all the partial
derivatives are continuous. Then,
∂P ∂Q
= , in D.
∂y ∂x

→
− →
−
Theorem 16. Let F = P i + Q j be a vector fields on an open simply-connected region
D. Supppose that all the partial derivatives are continuous and
∂P ∂Q
= in D.
∂y ∂x
Then F is conservative.

16.4 Green’s Theorem
Theorem 17. Let C be a positively oriented, piecewise-smooth, simple closed curve in the
plane and let D be the region bounded by C. Supppose that all the partial derivatives of P
and Q are continuous on an open region contains D, then
∂Q ∂P
P dx + Qdy = − dA
C D ∂x ∂y

Application: Line The formulas to find the area of D :
1
A= xdy = − ydxa = xdy − ydx.
C C 2 C

• See Examples 1,2,3

16.5 Curl and Divergence
→
− →
− →
−
Curl: If F = P i + Q j + R k is a vector field on R3 and partial derivatives of P , Q, and
R all exist, then the curl of F is the vector field defined by

curl F = ×F
∂R ∂Q →
− ∂R ∂P →
− ∂Q ∂P →
−
= − i − − j + − k
∂y ∂z ∂x ∂z ∂x ∂y
Theorem 18. If f is a function of three variables that has continuous second-order partial
derivatives, then
curl ( f ) = 0.
Theorem 19. If F is a vector field defined on all of R3 whose component functions have
continuous partial derivatives and curl F = 0, then F is a conservative vector field.
18

• See Examples 1,2,3
→
− →
− →
−
Divergence: If F = P i + Q j + R k is a vector field on R3 and partial derivatives of P ,
Q, and R all exist, then the divergence of F is the function of three variables defined by
divF = ·F
∂P ∂Q ∂R
= + +
∂x ∂y ∂z
→
− →
− →
−
Theorem 20. If F = P i + Q j + R k is a vector field defined on all of R3 and P , Q,
and R have continuous second-order partial derivatives, then
div curl F = 0.

• See Examples 4,5

19

Summary (chapter 1 chapter6)

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