Vector Calculus 2
There’s more to the subject of vector calculus than the material in chapter nine. There are a couple
of types of line integrals and there are some basic theorems that relate the integrals to the derivatives,
sort of like the fundamental theorem of calculus that relates the integral to the antiderivative in one
dimension.
13.1 Integrals
Recall the definition of the Riemann integral from section 1.6.
Z
b
a
dx f
(
x
) =
lim
Δ
x
k
→
0
N
X
k
=1
f
(
ξ
k
) Δ
x
k
(13
.
1)
This refers to a function of a single variable, integrated along that one dimension.
The basic idea is that you divide a complicated thing into little pieces to get an approximate
answer. Then you refine the pieces into still smaller ones to improve the answer and finally take the
limit as the approximation becomes perfect.
What is the length of a curve in the plane? Divide the curve into a lot of small pieces, then if the
pieces are small enough you can use the Pythagorean Theorem to estimate the length of each piece.
Δ
‘
k
=
p
(Δ
x
k
)
2
+ (Δ
y
k
)
2
Δ
x
k
Δ
y
k
The whole curve then has a length that you estimate to be the sum of all these intervals. Finally take
the limit to get the exact answer.
X
k
Δ
‘
k
=
X p
(Δ
x
k
)
2
+ (Δ
y
k
)
2
→
Z
d‘
=
Z
p
dx
2
+
dy
2
(13
.
2)
How do you actually
do
this? That will depend on the way that you use to describe the curve itself.
Start with the simplest method and assume that you have a parametric representation of the curve:
x
=
f
(
t
)
and
y
=
g
(
t
)
Then
dx
=
˙
f
(
t
)
dt
and
dy
=
˙
g
(
t
)
dt
, so
d‘
=
q
(
˙
f
(
t
)
dt
)
2
+
(
˙
g
(
t
)
dt
)
2
=
q
˙
f
(
t
)
2
+
˙
g
(
t
)
2
dt
(13
.
3)
and the integral for the length is
Z
d‘
=
Z
b
a
dt
q
˙
f
(
t
)
2
+
˙
g
(
t
)
2
where
a
and
b
are the limits on the parameter
t
. Think of this as
R
d‘
=
R
v dt
, where
v
is the speed.
James Nearing, University of Miami
1
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13—Vector Calculus 2
2
Do the simplest example first. What is the circumference of a circle? Use the parametrization
x
=
R
cos
φ,
y
=
R
sin
φ
then
d‘
=
p
(

R
sin
φ
)
2
+ (
R
cos
φ
)
2
dφ
=
R dφ
(13
.
4)
The circumference is then
R
d‘
=
R
2
π
0
R dφ
= 2
πR
.
An ellipse is a bit more of a challenge; see
problem
13.3
.
If the curve is expressed in polar coordinates you may find another formulation preferable, though
in essence it is the same. The Pythagorean Theorem is still applicable, but you have to see what it says
in these coordinates.
Δ
‘
k
=
p
(Δ
r
k
)
2
+ (
r
k
Δ
φ
k
)
2
r
k
Δ
φ
k
Δ
r
k
If this picture doesn’t seem to show much of a right triangle, remember there’s a limit involved, as
Δ
r
k
and
Δ
φ
k
approach zero this becomes more of a triangle. The integral for the length of a curve is then
Z
d‘
=
Z
p
dr
2
+
r
2
dφ
2
To actually do this integral you will pick a parameter to represent the curve, and that parameter may
even be
φ
itself. For an example, examine one loop of a logarithmic spiral:
r
=
r
0
e
kφ
.
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 Fall '08
 Nearing
 Vector Calculus, The American, Vector field, Stokes' theorem

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