8Some applications of integral theorems
IA Vector Calculus
8.1 Integral expressions for div and curl
We can use these theorems to come up with alternative definitions of the div and
curl. The advantage of these alternative definitions is that they do not require a
choice of coordinate axes. They also better describe how we should interpret div
and curl.
Gauss’ theorem for F in a small volume V containing r
0
gives
Z
∂V
F · dS =
Z
V
∇ · F dV ≈ (∇ · F)(r
0
) vol(V ).
We take the limit as V → 0 to obtain
Proposition.
(∇ · F)(r
0
) = lim
diam(V )→1
1
vol(V )
Z
∂V
F · dS,
where the limit is taken over volumes containing the point r
0
.
Similarly, Stokes’ theorem gives, for A a surface containing the point r
0
,
Z
∂A
F · dr =
Z
A
(∇ × F) · n dA ≈ n · (∇ × F)(r
0
) area(A).
So
Proposition.
n · (∇ × F)(r
0
) = lim
diam(A)→0
1
area(A)
Z
∂A
F · dr,
where the limit is taken over all surfaces A containing r
0
with normal n.
These are coordinate-independent definitions of div and curl.
Example. Suppose u is a velocity field of fluid flow. Then
Z
S
u · dS
is the rate of which fluid crosses
S
. Taking
V
to be the volume occupied by a
fixed quantity of fluid material, we have
˙
V =
Z
∂V
u · dS
Then, at r
0
,
∇ · u = lim
V →0
˙
V
V
,
the relative rate of change of volume. For example, if
u
(
r
) =
αr
(ie fluid flowing
out of origin), then ∇ · u = 3α, which increases at a constant rate everywhere.
Alternatively, take a planar area A to be a disc of radius a. Then
Z
∂A
u · dr =
Z
∂A
u · t ds = 2πa × average of u · t around the circumference.
(
u · t
is the component of
u
which is tangential to the boundary) We define the
quantity
ω =
1
a
× (average of u · t).
This is the local angular velocity of the current. As
a →
0,
1
a
→ ∞
, but the
average of
u · t
will also decrease since a smooth field is less “twirly” if you look
closer. So ω tends to some finite value as a → 0. We have
Z
∂A
u · dr = 2πa
2
ω.
Recall that
n · ∇ × u = lim
A→0
1
πa
2
Z
∂A
u · dr = 2ω,
ie twice the local angular velocity. For example, if you have a washing machine
rotating at a rate of ω , Then the velocity u = ω ×r. Then the curl is
∇ × (ω ×r) = 2ω,
which is twice the angular velocity.