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Consider a vector field , and a loop which lies in one plane.
The integral of around this loop is written
, where is a line element of the
loop. If is a conservative field then
and
for all loops. In general, for a nonconservative field,
.
For a small loop we expect
to be proportional to
the area of the loop. Moreover, for a fixed area loop we expect
to depend on the orientation of the loop.
One particular orientation will give the maximum value:
. If the loop subtends an angle with this optimum orientation
then we expect
. Let us introduce the vector field
whose magnitude is

(143) 
for the orientation giving . Here, is the area of the loop.
The direction of
is perpendicular to the plane of the loop,
when it is
in the
orientation giving , with the sense given by the righthand grip rule.
Figure 22:

Let us now express
in terms of the components of .
First, we shall
evaluate
around a small rectangle in the
 plane (see Fig. 22).
The contribution from sides 1 and 3 is

(144) 
The contribution from sides 2 and 4 is

(145) 
So, the total of all contributions gives

(146) 
where is the area of the loop.
Consider a nonrectangular (but still small) loop in the  plane.
We can divide it into rectangular
elements, and form
over all the resultant
loops. The interior
contributions cancel, so we are just left with the contribution from the outer loop.
Also, the area of the outer loop is the sum of all the areas of the inner loops.
We conclude that

(147) 
is valid for a small loop
of any shape in the  plane. Likewise, we can show that
if the loop is in the  plane then
and

(148) 
Finally, if the loop is in the  plane then
and

(149) 
Figure 23:

Imagine an arbitrary loop of vector area
. We
can construct this out of three loops in the , , and directions, as
indicated in Fig. 23.
If we form the line integral around all three loops then the interior contributions
cancel, and we are left with the line integral around the original loop. Thus,

(150) 
giving

(151) 
where

(152) 
Note that

(153) 
This demonstrates that
is a good vector field, since it is
the cross product of the operator (a good vector operator) and
the vector field .
Consider a solid body rotating about the axis. The angular velocity is given
by
, so the rotation velocity at
position is

(154) 
[see Eq. (43)].
Let us evaluate
on the axis of
rotation. The component is proportional to the
integral
around a loop in the  plane. This is
plainly zero. Likewise, the component is also zero. The component
is
around some loop in the  plane.
Consider a circular loop. We have
with
.
Here, is the radial distance from the rotation axis.
It follows that
, which
is independent of . So, on the axis,
.
Off the axis, at position , we can write

(155) 
The first part has the same curl as the velocity field on the axis, and the
second part has zero curl, since it is constant. Thus,
everywhere in the body. This allows us to
form a physical picture of
. If we imagine as the
velocity field of some fluid, then
at any given point is equal
to twice the local angular rotation velocity:
i.e., 2.
Hence, a vector field with
everywhere is said to
be irrotational.
Another important result of vector field theory is the curl theorem
or Stokes' theorem,

(156) 
for some (nonplanar) surface bounded by a rim . This theorem can easily
be proved by splitting the loop up into many small rectangular loops, and forming
the integral around all of the resultant loops. All of the contributions from the
interior loops cancel, leaving just the contribution from the outer rim.
Making use of Eq. (151) for each of the small loops, we can see that the contribution
from all of the loops is also equal to the integral of
across the whole surface. This proves the theorem.
One immediate consequence of of Stokes' theorem is that
is ``incompressible.'' Consider two surfaces, and , which share the
same rim. It is clear from Stokes' theorem that
is the same for both surfaces. Thus, it
follows that
for any closed surface.
However, we have from the divergence theorem that
for any volume. Hence,

(157) 
So,
is a solenoidal field.
We have seen that for a conservative field
for any loop. This is entirely equivalent to
.
However, the magnitude of
is
for some
particular loop. It is clear then that
for a conservative field. In other words,

(158) 
Thus, a conservative field is also an irrotational one.
Finally, it can be shown that

(159) 
where

(160) 
It should be emphasized, however, that the above result is only valid in
Cartesian coordinates.
Next: Summary
Up: Vectors
Previous: The Laplacian
Richard Fitzpatrick
20060202