Chapter
4
Vector Integral Calculus
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Table of Contents
1.
Line Integrals
2.
Path Independence of Line Integrals
3.
Double Integrals
4.
Green’s Theorem in the Plane
5.
Surfaces for Surface Integrals
6.
Surface Integrals
7.
Triple Integrals. Divergence Theorem of Gauss
8.
Applications of Divergence Theorem
9.
Stoke’s Theorem
Appendix
A1
Equation of Continuity
A2
Equation of Conservation
Vector Integral Calculus
1. Line Integrals
Vector Integral Calculus
1.
Line Integrals
Let
C
be a curve, there are several types of line integrals.
Definition:
Line Integral
Let
( ,
, )
f
x y z
be a continuous function defined at all point of
C
.
The line integral of
( ,
, )
f
x y z
over the curve
C
is defined as
0
1
( ,
, )
lim
(
)
k
n
k
k
C
s
k
f x y z ds
f P
s
where we partition the curve into subarcs with
k
P
(any point in the
k
th subarc) and
k
s
(length of the
k
th subarc).
C
is called the path of integration that initial point
A
(initial point) and
B
(terminal point).
C
is smooth curve if
C
has unique tangent at each of points varies continuously.
The direction from
A
to
B
is called the
positive
direction on
C
(counterclockwise).
Definition:
Line Integral
Let
1
(
,
,
)
i
i
i
F x
y
z
,
2
(
,
,
)
i
i
i
F
x
y
z
and
3
(
,
,
)
i
i
i
F
x
y
z
be continuous differentiable functions defined
on
C
.
The line integral is
1
2
3
1
2
3
0
1
lim
(
,
,
)
(
,
,
)
(
,
,
)
k
C
n
k
k
k
k
k
k
k
k
k
k
k
k
s
k
F dx
F dy
F dz
F x
y
z
x
F
x
y
z
y
F
x
y
z
z
where we partition the curve
C
with
(
,
,
)
k
k
k
x
y
z
(any point in the
k
th subarc) and
,
,
k
k
k
x
y
z
(the usual difference in
,
,
x y z
) .
How to compute line integral
To integrate a continuous function
( ,
, )
f
x y z
over a curve
C
:
1.
Parameterize the curve
C
:
( )
( )
( )
( )
t
x t
y t
z t
r
i
j
k
2.
Change variables (from
s
to
t
or from
( ,
, )
x y z
to
t
)
( ,
, )
( ( ),
( ), ( )) 
( ) 
b
C
a
f
x y z ds
f
x t
y t
z t
t
dt
r
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Vector Integral Calculus
1. Line Integrals
Recall:
The arc length of a curve
C
from point
A
to
( ,
, )
x y z
is given by
2
2
2
/
/
/
b
b
a
a
s
ds
dx
dt
dy
dt
dz
dt
dt
1.1.
Vector Form of Line Integrals
Definition:
Vector Line Integral
Let
C
be a curve given by a position vector
( )
[ ( ),
( ),
( )]
( )
( )
( )
t
x t
y t
z t
x t
y t
z t
r
i
j
k
and let
( ,
, )
( )
x y z
F
F r
be a vector field defined on
C
.
We define the line integral
a
( )
( ( ))
b
C
d
d
t
dt
dt
r
F r
r
F r
as follows


0
1
( )
lim
(
,
,
)
n
i
i
i
i
C
i
d
x
y
z
r
F r
r
F
r
where
1
( )
(
)
i
i
i
t
t
r
r
r
or
i
i
i
i
x
y
z
r
i
j
k
.
※
work done by force
F
in a displacement along
C
(work integral)
▶
Closed Path
―
A
and
B
may coinside.
▶
Piecewise Continuous
―
it consist of finitely many smooth curves.
▶
Smooth curve
C
―
C
has a unique continuous tangent or
( )
t
r
is differentiable and
'( )
/
t
d
dt
r
r
is continuous on
C