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 xyz
be a continuous function defined at all point of
C
.
The line integral of (, ,)
over the curve
C
is defined as
0
1
(, ,)
l
im
( )
k
n
kk
C
s
k
f xyzd
s
fP s
where we partition the curve into sub-arcs with
k
P
(any point in the
k
-th sub-arc) and
k
s
(length of the
k
-th sub-arc).
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
(, ,)
iii
Fxyz
,
2
and
3
be continuous differentiable functions defined
on
C
.
The line integral is
123
12
3
0
1
lim
(
,
,
)
(
,
,
)
(
,
,
)
k
C
n
kkk k
s
k
Fdx Fdy Fdz
Fxyz x Fxyz y Fxyz z
where we partition the curve
C
with (, ,)
kkk
xyz
(any point in the
k
-th sub-arc) and
,
x
yz
(the usual difference in ,,
) .
How to compute line integral
To integrate a continuous function (, ,)
over a curve
C
:
1.
Parameterize the curve
C
: ()
()
tx
ty
tz
t
ri
j
k
2.
Change variables (from
s
to
t
or from (, ,)
to
t
)
( , , )
( ( ), ( ), ( )) |
( ) |
b
Ca
s
f xt yt zt
t d
t
r