Chapter 12:
VECTORS
1. Geometry: Let P1 (x1 , y1 , z1 ) and P2 (x2 , y2 , z2 ) be points in 3-space:
A. Distance Formula: d(P1 , P2 ) =
p
(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
B. Midpoint Formula: The midpoint of the line segment joining P1 and P2 is the point
x1 + x2 y1 + y2 z1 + z2
,
,
P
2
2
2
C. Equation for the sphere of radius r and center P (a, b, c):
(x − a2 + (y − b)2 + (z − c)2 = r2
2.
Vectors: A vector a in n-dimensional space is an ordered n-tuple of real numbers. In particular, a
vector a in 3-space is an ordered triple of numbers: a = (a1 , a2 , a3 ); a vector a in the plane (2-space) is
an ordered pair of numbers: a = (a1 , a2 ). The vector 0 = (0, 0, 0) is the zero vector
Let a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) be vectors in 3-space and let α be a real number (scalar).
A. Equality: a = b iff a1 = b1 , a2 = b2 , a3 = b3
B. Vector Addition: a + b = (a1 + b1 , a2 + b2 , a3 + b3 )
C. Multiplication by a Scalar: α a = (α a1 , α a2 , α a3 )
NOTE: a and b are parallel iff a = λb for some number λ.
p
D. Magnitude (Norm): kak = a21 + a22 + a23 ; a is a unit vector if kak = 1. If a 6= 0 , then
a
is a unit vector in the direction of a.
ua =
kak
(i) kak ≥ 0;
kak = 0 iff a = 0 .
(ii) kαak = | α| kak.
(iii) ka + bk ≤ kak + kbk.
E. Unit Coordinate Vectors: i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)
F. i, j, k-Representation: a = (a1 , a2 , a3 ) = a1 i + a2 j + a3 k.
1
3. Dot Product:
a · b = a1 b1 + a2 b2 + a3 b3 .
The dot product of two n-dimensional vectors is defined similarly.
A. Properties:
(i) a · a = kak2
(ii) a · 0 = 0
(iii) a · = b · a
(iv) (αa) · (βb) = αβ (a · b)
(v) a · (b + c) = a · b + a · c
B. Geometric Interpretation:
a · b = kak kbk cos θ where θ is the angle between a and
b, 0 ≤ θ ≤ π. a is perpendicular to b (a ⊥ b) iff a · b = 0.
C. Component & Projection of a on b:
a· b
a· b b
= a · ub
·
= (a · ub ) ub
comp b a =
proj b a =
kbk
kbk kbk
D. Direction Angles; Direction Cosines: a · i = kak cos α, a · j = kak cos β, a · k = kak cos γ.
The angles α, β, γ are called the direction angles of a; cos α, cos β, cos γ are called the direction
cosines of a.
4. Cross Product: This product is restricted to vectors in 3-space. Let a and b be vectors in 3-space
such that a 6= λb. The cross product of a and b, denoted by a × b is the vector defined as follows:
(1) a × b is perpendicular to the plane determined by a and b.
(2) a, b and a × b (in this order) form a right-handed triple.
(3) ka × bk = kak kbk sin θ where θ is the angle between a and b.
If a = λb for some number λ; that is, if a and b are parallel, then a × b = 0 .
A. Properties:
(i) a × b = −b × a.
(ii) (αa) × (βb) = α β (a × b).
(iii) a × (b + c) = a × b + a × c.
B. Components of a × b: Let a = (a1 , a2 , a3 ) , b = (b1 , b2 , b3 )
2
i
a × b = a1
b1
j
a2
b2
k a3 = (a2 b3 − a3 b2 ) i − (a1 b3 − a3 b1 ) j + (a1 b2 − a2 b1 ) k
b3 C. Triple Scalar Product: Let a = (a1 , a2 , a3 ) ,
a1
(a × b) · c = b1
c1
b = (b1 , b2 , b3 ), c = (c1 , c2 , c3 ). Then
a2 a3 b2 b3 c2 c3 The volume of the parallelepiped having a, b and c as sides is given by:
| (a × b) · c |
5. Lines: There is one and only one line ` passing through a given point P0 : (x0 , y0 , z0 ) parallel to a
given vector d = (d1 , d2 , d3 ). The vector d is called a direction vector for `; the numbers d1 , d2 , d3 are
direction numbers for `.
A. Equations for `:
Vector Equation of ` :
r(t) = r0 + t d = (x0 + td1 ) i + (y0 + td2 ) j + (z0 + td3 ) k
Scalar Parametric Equations of `:
x(t) = x0 + t d1 ,
y(t) = y0 + t d2 ,
z(t) = z0 + t d3
Symmetric Equations of `:
x − x0
y − y0
z − z0
=
=
d1
d2
d3
B. Two Lines in Space: Let ` and L be two lines in space, and let d and D be corresponding
direction vectors. ` and L are either parallel or coincident if d = λD; ` and L intersect in a
point or are skew if d 6= λD.
C. Angle φ between ` and L:
cos φ =
|d · D|
kdk kDk
D. Distance From a Point P1 : (x1 , y1 , z1 ) to the Line `:
−−→
kP0 P1 × dk
.
d(P1 , `) =
kdk
6. Planes: There is one and only one plane P passing through a given point P0 : (x0 , y0 , z0 ) perpendicular
to a given vector N = A i + b j + C k. The vector N is called a normal vector to the plane P.
3
A. Equations for P:
”Standard Form:”
A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0
Removing the parentheses yields the equation
Ax + By + Cz + D = 0,
D = −Ax0 − By0 − Cz0
which can also be written in the form
Ax + By + Cz = E.
B. Two Planes in Space: Let P1 and P2 be two planes in space, and let N1 and N2 be
corresponding normal vectors. P1 and P2 are either parallel or coincident if N1 = λN2 ; P1 and
P2 intersect in a line if N1 6= λN2 .
C. Angle θ between P1 and P2 :
cos θ =
| N1 · N2 |
kN1 k kN2k
C. Distance From a Point P1 : (x1 , y1 , z1 ) to the Plane P:
d(P1 , P) =
kAx1 + By1 + Cz1 + Dk
√
A2 + B 2 + C 2
4
Chapter 13:
VECTOR CALCULUS
VECTOR FUNCTIONS:
Let f1 (t), f2 (t), f3 (t) be functions defined on some t-interval I. For
each t ∈ I, form the vector
f (t) = f1 (t) i + f2 (t) j + f3 (t) k.
f is called a vector-valued function or, more simply, a vector function. Note: there is nothing special
about 3 dimensions here; we can have vector functions in any number of coordinates. In particular,
we will also see many examples in two dimensions.
1. CALCULUS:
Let f (t) = f1 (t) i + f2 (t) j + f3 (t) k.
A. Limits: Let c ∈ I. Then
lim f (t) = L = L1 i + L2 j + L3 k if and only if
lim kf (t) − Mk = 0
t→c
if and only if
lim f1 (t) = L1 ,
t→c
lim f2 (t) = L2 ,
t→c
lim f3 (t) = L3
t→c
t→c
The usual limit theorems hold: limit of a sum, difference, etc.
B. Derivatives:
f is differentiable at t if and only if
lim
h→0
1
[f (t + h) − f (t)]
h
exists.
If the limit exists, it is called the derivative of f at t and is denoted f 0 (t). Because of the
properties of limits indicated in (A),
f 0 (t) = f10 (t) i + f20 (t) j + f30 (t) k.
C. Integrals:
2.
Z
GEOMETRY:
f (t) dt =
Z
f1 (t) dt i +
Z
f2 (t) dt j +
Z
f1 (t) dt k
Let x = x(t), y = y(t), and z = z(t) be differentiable functions on some
t-interval I and let r(t) = x(t) i + y(t) j + z(t) k. The tip of the vector r traces out a curve C in
space. The equations:
x = x(t), y = y(t), z = z(t)
are parametric equations for C. Also, C is an oriented curve ; the ”positive direction” on I induces
a positive direction on C.
A. Tangent Vector:
The vector
r0 (t) = x0 (t) i + y 0 (t) j + z 0 (t) k,
if not 0 , is a direction vector for the line tangent to C at the point (x(t), y(t), z(t)) on C; r0 (t)
points in the direction of increasing t.
1
If r0 (t) 6= 0 , then
B. Unit Tangent Vector:
T(t) =
r0 (t)
kr0 (t)k
is the unit tangent vector.
If T0 (t) 6= 0 , then
C. Principal Normal Vector:
N(t) =
T ⊥ N.
is the principal normal vector;
D. Osculating Plane:
T0 (t)
kT0 (t)k
The plane determined by T and N is called the osculating plane. In
particular, choose c ∈ I. Then P (x(c), y(c), z(c)) is a point on the curve C and T(c) × N(c)
is a normal vector for the osculating plane at P . This is the information needed to write an
equation for the osculating plane.
3.
MECHANICS:
Let r(t) = x(t) i + y(t) j + z(t) k denote the position at time t of an object.
As t ranges over the interval I, the object moves along the curve C.
A. Velocity Vector:
The vector
v(t) = r0 (t) = x0 (t) i + y 0 (t) j + z 0 (t) k
is called the velocity vector and
p
kr0 (t)k = [x0 (t)]2 + [y 0 (t)]2 + [z 0 (t)]2
is the speed of the object at time t.
B. Acceleration Vector:
The vector
a(t) = v0 (t) = r00 (t) = x00 (t) i + y 00 (t) j + z 00 (t) k
is called the acceleration vector.
4. ARC LENGTH/DISTANCE:
Let a, b ∈ I, a < b. The length of the curve C for a ≤ t ≤ b
is given by:
L(C) =
Z
b
Z
p
[x0 (t)]2 + [y 0 (t)]2 + [z 0 (t)]2 dt =
a
Distance:
b
kr0 (t)k dt
a
The distance s(t) traveled by a particle moving along C over the time interval
[a, t] is given by:
Z tp
s(t) =
[x0 (u)]2 + [y 0 (u)]2 + [z 0 (u)]2 du
a
2
The derivative of distance with respect to time, ds/dt, is
ds p 0 2
= [x (t)] + [y 0 (t)]2 + [z 0 (t)]2 = kr0 (t)k,
dt
•
see Section 5.2
and this is the speed of the object as noted above.
5. CURVATURE:
The curvature of a curve is a measure of the rate at which the curve is
”curving.” The curvature of C is the magnitude of the change of the unit tangent vector with respect
to arc length:
T kT/dtk
kT0 (t)k
κ=
ds = ds/dt = kr0 (t)k
Special Cases:
(1) C : y = f (x) the graph of a function:
κ=
|y 00 (t)|
(1 + [y 0 ]2 )
3/2
(2) C : x = x(t), y = y(t) a plane curve defined parametrically”
κ=
|x0 (t)y 00 (t) − y 0 (t)x00 (t)|
([x0 ]2 + [y 0 ]2 )3/2
6. TANGENTIAL AND NORMAL COMPONENTS OF ACCELERATION:
The ac-
celeration vector
a(t) = a(t) = x00 (t) i + y 00 (t) j + z 00 (t) k
is a linear combination of the unit tangent and principal normal vectors; i.e., the acceleration vector
lies in the osculating plane:
a(t) = aT T + aN N
The coefficients aT and aN are called the tangential and normal components of acceleration; aT
and aN are given by:
aT =
d2 s
,
dt2
aN = κ
3
ds
dt
2
SUMMARY: CHAPTERS 14 and 15
1. Functions of Several Variables: Let z = f (x, y) or w = f (x, y, z).
Domain D: If the domain of f is not given explicitly, or implicitly by an application,
then, by convention, the domain is the set of all points x [x = (x, y) or x = (x, y, z)]
such that f (x) is a real number.
Range: The range of f is the set of values f (x), x ∈ D.
Graph: Let z = f (x, y), (x, y) ∈ D. The graph of f is the set of all points
(x, y, z) = (x, y, f (x, y)) in space. For our purposes, the graph of z = f (x, y) is a
surface in space.
The graph of w = f (x, y, z) is a “hypersurface” in 4-dimensional space.
Level curves and level surfaces: Given z = f (x, y). The plane curves f (x, y) =
C, C constant are called the level curves of f . Let w = F (x, y, z). The surfaces in
3-space F (x, y, z) = C are called the level surfaces of F .
2. Limits: Given z = f (x, y) or w = f (x, y, z).
Let x0 = (x0, y0) or (x0, y0, z0) and x = (x, y) or (x, y, z). Assume that f is
defined in some neighborhood of the point x0 except, possibly, at x0 itself.
lim f (x) = L
x→x0
if for each > 0 there exists a δ > 0 such that
|f (x) − L| < whenever
kx − x0k < δ.
3. Partial Derivatives: Given a function z = f (x, y). Let x = (x, y).
The partial derivative of f with respect to x at the point x is given by
fx = lim
h→0
f (x + h, y) − f (x, y)
f (x + h i) − f (x)
= lim
h→0
h
h
(provided the limit exists) .
The partial derivative of f with respect to y at the point x is given by
fy = lim
h→0
f (x, y + h) − f (x, y)
f (x + h j) − f (x)
= lim
h→0
h
h
(provided the limit exists) .
Corresponding definitions hold for the function w = F (x, y, z). For example:
The partial derivative of F with respect to z at the point x = (x, y, z) is given by
1
Fz = lim
h→0
F (x, y, z + h) − F (x, y, z)
F (x + h k) − F (x)
= lim
h→0
h
h
Notations:
fx =
∂f
∂z
=
,
∂x
∂x
fy =
(provided the limit exists) .
∂f
∂z
=
, etc.
∂y
∂y
Higher Derivatives:
fxx =
∂ 2f
∂ (∂f /∂x)
=
,
∂x2
∂x
fxy =
∂ 2f
∂ (∂f /∂x)
=
,
∂y ∂x
∂y
fyy =
∂ 2f
∂ (∂f /∂y)
=
,
2
∂y
∂y
fyx =
∂ 2f
∂ (∂f /∂y)
=
.
∂x ∂y
∂x
THEOREM: If the first partials and the “mixed” second partials are continuous on
D, then fxy = fyx .
4. Gradient: Given z = f (x, y); w = F (x, y, z)
Gradient of f : ∇f = fx i + fy j
gradient of F : ∇F = Fx i + Fy j + Fz k
∇f is normal to the level curves of f ; ∇F is normal to the level surfaces of F .
Directional derivatives: Let u = u1 i + u2 j be a unit vector and let x = (x, y).
The directional derivative of f at x in the direction u is given by:
fu0 = lim
h→0
f (x + h u) − f (x)
h
provided the limit exists
Similarly, if u = u1 i+u2 j+u3 k is a unit vector and x = (x, y, z), then the directional
derivative of F at x in the direction u is
Fu0 = lim
h→0
F (x + h u) − F (x)
h
provided the limit exists
Calculation of directional derivatives:
fu0 = ∇f (x)·u;
Fu0 = ∇F (x)·u.
Tangent planes and normal lines: Equations for the tangent plane and normal
line to the surface z = f (x, y) at the point (x0, y0, z0), [z0 = f (x0 , y0)] are given by:
tangent plane: fx (x0 , y0)(x − x0 ) + fy (x0, y0)(y − y0 ) − (z − z0 ) = 0;
2
normal line: x = x0 + fx (x0, y0) t, y = y0 + fy (x0 , y0) t, z = z0 − t
Equations for the tangent plane and normal line to the level surface F (x, y, z) = C at
the point P0 (x0 , y0, z0) are given by:
tangent plane: Fx (P0 )(x − x0 ) + Fy (P0)(y − y0 ) + Fz (P0 )(z − z0 ) = 0;
normal line: x = x0 + Fx (P0) t, y = y0 + Fy (P0 )t, z = z0 + Fz (P0) t.
5. Chain Rules: Given z = f (x, y); w = F (x, y, z)
a. If x = x(t) and y = y(t), then:
df
∂f dx ∂f dy
=
+
.
dt
∂x dt
∂y dt
dF
∂F dx ∂F dy ∂F dz
=
+
+
.
dt
∂x dt
∂y dt
∂z dt
∂f
∂f ∂x ∂f ∂y
=
+
.
ds
∂x ∂t
∂y ∂t
If x = x(t), y = y(t) and z = z(t), then:
b. If x = x(s, t) and y = y(s, t), then:
If x = x(s, t), y = y(s, t) and z = z(s, t), then:
Similarly for
∂F
∂F ∂x ∂F ∂y ∂F ∂z
=
+
+
.
∂t
∂x ∂t ∂y ∂t ∂z ∂t
∂f ∂F
,
.
∂s ∂s
6. Extreme Values: Let f be a function of several variables [z = f (x, y), w = f (x, y, z)], and
let x0 be an interior point of the domain of f .
a. f has a local maximum at x0 if f (x0) ≥ f (x) for all x in some neighborhood
of x0.
b. f has a local minimum at x0 if f (x0) ≤ f (x) for all x in some neighborhood
of x0.
c. Theorem: If f has a local extreme value at x0, then either
∇f (x0) = 0
or
∇f (x0) does not exist.
A point x0 at which either of these conditions holds is called a critical point of f ; a
point x0 at ∇f (x0) = 0 is called a stationary point of f .
Second Partials Test: Let z = f (x, y) have continuous first and second partial
derivatives in a neighborhood of a stationary point x0. Set
A=
∂ 2f
(x0),
∂x2
B=
∂ 2f
(x0),
∂x ∂y
and set D = AC − B 2 .
3
C=
∂ 2f
(x0)
∂y 2
1. If D < 0, then x0 is a saddle point of f .
2. If D > 0, then f has:
a local minimum at x0 if A > 0,
a local maximum at x0 if A < 0.
3. ???? if D = 0.
7. Lagrange Multipliers:
If x0 maximizes or minimizes f subject to the side condition g(x) = 0, then there
exists a scalar λ such that
∇f (x0) = λ ∇g(x0)
(provided ∇g(x0) 6= 0).
To find the maxima/minima of z = f (x, y) subject to the side condition g(x, y) = 0,
solve the system of equations:
fx (x, y) = λ gx(x, y)
fy (x, y) = λ gy (x, y)
g(x, y) = 0
To find the maxima/minima of w = F (x, y, z) subject to the side condition G(x, y, z) =
0, solve the system of equations:
Fx (x, y, z)
Fy (x, y, z)
Fz (x, y, z)
G(x, y, z)
=
=
=
=
λ Gx(x, y, z)
λ Gy (x, y, z)
λ Gz (x, y, z)
0
8. Reconstructing a Function from its Gradient:
Let P = P (x, y) and Q = Q(x, y) be continuously differentiable functions on a simply
connected open region Ω. The vector function R(x, y) = P (x, y) i + Q(x, y) j is the
gradient of some function z = f (x, y) on Ω if and only if
∂P
∂Q
=
.
∂y
∂x
If R is the gradient of a function f , then
f (x, y) =
Z
P (x, y) dx + φ(y)
where φ is a function of y which is to be determined so that ∂f /∂y = Q(x, y).
Alternatively,
f (x, y) =
Z
Q(x, y) dy + ψ(y)
where ψ is a function of x which is to be determined so that ∂f /∂x = P (x, y).
4
SUMMARY: CHAPTERS 16 and 17
CHAPTER 16 MULTIPLE INTEGRALS
I. DOUBLE INTEGRALS
a.
Definition: Let f = f (x, y) be continuous on the rectangle R : a ≤ x ≤ b, c ≤ y ≤ d.
Let P be a partition of R and let mij and Mij be the minimum and maximum values
of f on the i, j sub-rectangle Rij . Then
n X
m
X
(i) Lower sum: Lf (P) =
mij ∆xi ∆yj .
i=1 j=1
n X
m
X
(ii) Upper sum: Uf (P) =
Mij ∆xi ∆yj .
i=1 j=1
n X
m
X
(iii) Riemann sum: Sf (P) =
f (x∗i , yj∗ )∆xi ∆yj where (x∗i , yj∗ ) is a point in Rij .
i=1 j=1
The double integral of f over R is the unique number I that satisfies
Lf (P) ≤ I ≤ Uf (P)
Notation: I =
Z Z
for all partitions P.
f (x, y) dxdy
R
Let Ω be an arbitrary closed bounded region in the plane. Then
Z Z
Z Z
f (x, y) dxdy =
Ω
F (x, y) dxdy
R
where R is a rectangle that contains Ω, and F (x, y) = f (x, y) on Ω and F (x, y) = 0
on R − Ω.
b.
Repeated Integrals: If the region Ω is given by: a ≤ x ≤ b, φ1 (x) ≤ y ≤ φ2 (x) (Type
I region), then
Z Z
f (x, y) dxdy =
Z
Ω
b Z φ2 (x)
f (x, y) dy dx
φ1 (x)
a
If the region Ω is given by: c ≤ y ≤ d, ψ1 (y) ≤ x ≤ ψ2 (y) (Type II region), then
Z Z
f (x, y) dxdy =
Ω
c. Polar Coordinates:
Z Z
Z
d
Z
Ω
Z Z
f (r cos θ, r sin θ) r drdθ.
Ω
1
f (x, y) dx dy
ψ1 (y)
c
f (x, y) dxdy =
ψ2 (y)
d.
Applications:
(i) Volume: If f (x, y) ≥ 0 on Ω, then V =
Z Z
f (x, y) dxdy is the volume of the
Ω
solid “cylinder” that has the surface z = f (x, y) as its top, Ω as its base, and vertical
sides.
Z Z
(ii) Area:
1 dxdy = area of Ω.
Ω
(iii) Mass of a Plate: If the density of a “plate” at a point (x, y) in the closed, bounded
region Ω is given by a continuous function λ(x, y), then the mass of the plate is
M=
Z Z
λ(x, y) dxdy
Ω
(iv) Center of Mass of a Plate: Let the continuous function λ(x, y) be the density
function of a plate. Then the coordinates (xM , yM ) of the center of mass of the plate
are given by:
Z Z
Ω
xM =
Z Z
x λ(x, y) dxdy
,
M
where M is the mass of the plate.
y λ(x, y) dxdy
Ω
yM =
M
II. TRIPLE INTEGRALS
a.
Definition: Let f = f (x, y, z) be continuous on the “box”
T : a1 ≤ x ≤ a2 , b1 ≤ y ≤ b2 , c1 ≤ z ≤ c2 .
Let P be a partition of T , and let mijk and Mijk be the minimum and maximum
values of f on the ijk sub-box Rijk . Then
(i) Lower sum: Lf (P) =
n X
m X
l
X
mijk ∆xi ∆yj ∆zk .
(ii) Upper sum: Uf (P) =
i=1 j=1 k=1
n X
m X
l
X
Mijk ∆xi ∆yj ∆zk .
i=1 j=1 k=1
n X
m X
l
X
(iii) Riemann sum: Sf (P) =
f (x∗i , yj∗ , zk∗ )∆xi ∆yj ∆zk where (x∗i , yj∗ , zk∗ ) is
i=1 j=1 k=1
a point in Rijk .
The triple integral of f over T is the unique number I that satisfies
Lf (P) ≤ I ≤ Uf (P)
Notation: I =
Z Z Z
for all partitions P.
f (x, y, z) dxdydz
T
Let Ω be an arbitrary closed bounded region in space. Then
Z Z Z
f (x, y, z) dxdyz =
Ω
Z Z Z
F (x, y, z) dxdydz
R
where T is a “box” that contains Ω, and F (x, y, z) = f (x, y, z) on Ω and F (x, y) = 0
on T − Ω.
2
b.
Repeated Integrals: If the region Ω is given by:
a ≤ x ≤ b, φ1 (x) ≤ y ≤ φ2 (x), ψ1 (x, y) ≤ z ≤ ψ2 (x, y)
(Type I region),
then
Z Z Z
f (x, y, z) dxdydz =
Z
b
a
Ω
Z
φ2 (x)
φ1 (x)
Z
ψ2 (x,y)
f (x, y, z) dz dy dx
ψ1 (x,y)
Note: There are five more types of special regions.
c. Cylindrical Coordinates:
Z Z Z
f (x, y, z) dxdydz =
Z Z Z
Ω
f (r cos θ, r sin θ, z) r drdθdz.
Ω
CHAPTER 17. LINE INTEGRALS
Given a vector field
F(x, y) = P (x, y) i + Q(x, y) j
and a smooth (or piecewise smooth) curve C:
C : x = x(u), y = y(u),
a≤u≤b
C : r(u) = x(u) i + y(u) j
(parametric form)
a≤u≤b
(vector form)
Or, in three dimensions, a vector field
F(x, y, z) = P (x, y, z) i + Q(x, y, z) j + R(x, y, z) bf k
and a smooth (or piecewise smooth) curve C:
C : x = x(u), y = y(u), z = z(u),
a≤u≤b
C : r(u) = x(u) i + y(u) j + z(u) k
(parametric form)
a≤u≤b
(vector form)
DEFINITION: The line integral of F over C is the number given by
Z
F(r) · dr =
C
Z
b
a
3
F[r(u)] · r0 (u) du
Alternative notations:
Z
F(r) · dr =
C
Z
P (x, y) dx + Q(x, y) dy =
C
Z
FT ds
(2 dimensions)
C
or
Z
F(r) · dr =
Z
C
P (x, y, z) dx + Q(x, y, z) dy + R(x, y, z) dz =
C
Z
FT ds
(3 dimensions)
C
where FT is the component of F on the unit tangent vector T .
THEOREM:
Line integrals are invariant under orientation-preserving changes of parameter.
THEOREM:
Reversing the orientation of C changes the sign of the integral:
Z
F(r) · dr = −
−C
c.f.
Z
a
f (x) dx = −
b
Z
Z
F(r) · dr.
C
b
f (x) dx.
a
FUNDAMENTAL THEOREM OF LINE INTEGRALS:
Given a curve C : r(u), a ≤ u ≤ b and a vector field F. If F = ∇f for some function
f (x, y), then
Z
F(r) · dr = f (B) − f (A)
C
where A = r(a) and B = r(b).
DEFINITION:
The curve C is closed if r(a) = r(b).
COROLLARY 1. If F is the gradient of some function f and the curve C is closed, then
Z
F(r) · dr = 0
C
COROLLARY 2. (Independence of Path) If F is the gradient of some function f and if
C1 and C2 are any two curves which begin at A and end at B, then
Z
F(r) · dr =
C1
Z
F(r) · dr
C2
DEFINITION: The curve C is a simple closed curve if it is closed and r(t1 ) 6= r(t2 ) for
all a < t1 < t2 < b. The positive direction on C is counterclockwise. A simple closed curve is
also called a Jordan curve . The region enclosed by a simple closed curve is called a Jordan
region.
4
GREEN’S THEOREM: Given a simple closed curve C oriented in the counterclockwise
direction, and a vector field F(x, y) = P (x, y) i + Q(x, y) j.
I
P (x, y) dx + Q(x, y) dy =
Z Z C
Ω
∂Q ∂P
−
∂x
∂y
dx dy
where Ω is the Jordan region enclosed by C.
COROLLARY TO GREEN’S THEOREM
If C is a simple closed curve, and if
∂Q
∂P
=
; that is, if F = P (x, y) i + Q(x, y) j is a gradient, then,
∂y
∂x
I
P (x, y) dx + Q(x, y) dy = 0.
C
AREA OF Ω USING GREEN’S THEOREM:
region Ω:
Area of Ω =
Z Z
1 dx dy =
Ω
I
−y dx =
C
C a simple closed curve enclosing the
I
x dy =
C
5
1
2
I
−y dx + x dy.
C