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Curl

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 non-conservative 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 right-hand grip rule.

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 non-rectangular (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)

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 (non-planar) 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 2006-02-02