2Curves and Line
IA Vector Calculus
2.2 Line integrals of vector fields
Definition
(Line integral)
.
The line integral of a smooth vector field
F
(
r
) along
a path
C
parametrised by
r
(
u
) along the direction (orientation)
r
(
α
)
→ r
(
β
) is
Z
C
F(r) · dr =
Z
β
α
F(r(u)) · r
0
(u) du.
We say d
r
=
r
0
(
u
)d
u
is the line element on
C
. Note that the upper and lower
limits of the integral are the end point and start point respectively, and
β
is not
necessarily larger than α.
For example, we may be moving a particle from
a
to
b
along a curve
C
under a force field
F
. Then we may divide the curve into many small segments
δr
. Then for each segment, the force experienced is
F
(
r
) and the work done is
F(r) · δr. Then the total work done across the curve is
W =
Z
C
F(r) · dr.
Example.
Take
F
(
r
) = (
xe
y
, z
2
, xy
) and we want to find the line integral from
a = (0, 0, 0) to b = (1, 1, 1).
a
b
C
1
C
2
We first integrate along the curve
C
1
:
r
(
u
) = (
u, u
2
, u
3
). Then
r
0
(
u
) =
(1, 2u, 3u
2
), and F(r(u)) = (ue
u
2
, u
6
, u
3
). So
Z
C
1
F · dr =
Z
1
0
F · r
0
(u) du
=
Z
1
0
ue
u
2
+ 2u
7
+ 3u
5
du
=
e
2
−
1
2
+
1
4
+
1
2
=
e
2
+
1
4
Now we try to integrate along another curve
C
2
:
r
(
t
) = (
t, t, t
). So
r
0
(
t
) =
(1, 1, 1).
Z
C
2
F · dr =
Z
F · r
0
(t)dt
=
Z
1
0
te
t
+ 2t
2
dt
=
5
3
.
We see that the line integral depends on the curve C in general, not just a, b.
We can also use the arclength
s
as the parameter. Since d
r
=
t
d
s
, with
t
being the unit tangent vector, we have
Z
C
F · dr =
Z
C
F · t ds.
Note that we do not necessarily have to integrate
F ·t
with respect to
s
. We can
also integrate a scalar function as a function of
s
,
R
C
f
(
s
) d
s
. By convention,
this is calculated in the direction of increasing s. In particular, we have
Z
C
1 ds = length of C.
Definition
(Closed curve)
.
A closed curve is a curve with the same start and
end point. The line integral along a closed curve is (sometimes) written as
H
and is (sometimes) called the circulation of F around C.
Sometimes we are not that lucky and our curve is not smooth. For example,
the graph of an absolute value function is not smooth. However, often we can
break it apart into many smaller segments, each of which is smooth. Alternatively,
we can write the curve as a sum of smooth curves. We call these piecewise smooth
curves.
Definition
(Piecewise smooth curve)
.
A piecewise smooth curve is a curve
C
=
C
1
+
C
2
+
···
+
C
n
with all
C
i
smooth with regular parametrisations. The
line integral over a piecewise smooth C is
Z
C
F · dr =
Z
C
1
F · dr +
Z
C
2
F · dr + ··· +
Z
C
n
F · dr.
Example.
Take the example above, and let
C
3
=
−C
2
. Then
C
=
C
1
+
C
3
is
piecewise smooth but not smooth. Then
I
C
F · dr =
Z
C
1
F · dr +
Z
C
3
F · dr
=
e
2
+
1
4
−
5
3
= −
17
12
+
e
2
.
a
b
C
1
C
3