Curl

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

(A.150) |

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 a right-hand circulation rule.

Let us now express in terms of the components of . First, we shall evaluate around a small rectangle in the - plane, as shown in Figure A.25. The contribution from sides 1 and 3 is

(A.151) |

The contribution from sides 2 and 4 is

(A.152) |

So, the total of all contributions gives

(A.153) |

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

(A.154) |

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

(A.155) |

Finally, if the loop is in the - plane then and

(A.156) |

Imagine an arbitrary loop of vector area . We can construct this out of three vector areas, , , and , directed in the -, -, and -directions, respectively, as indicated in Figure A.26. 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,

(A.157) |

giving

where

(A.159) |

and is the angle subtended between the directions of and . Note that

(A.160) |

This demonstrates that is a good vector field, because 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

(A.161) |

[See Equation (A.52).] 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 perpendicular distance from the rotation axis. It follows that , which is independent of . So, on the axis, . Off the axis, at position , we can write

(A.162) |

The first part has the same curl as the velocity field on the axis, and the second part has zero curl, because 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: that is, 2 . Hence, a vector field with everywhere is said to be

Another important result of vector field theory is the *curl theorem*:

(A.163) |

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 Equation (A.158) 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 the curl theorem is that is ``incompressible.'' Consider any two surfaces, and , that share the same rim. (See Figure A.23.) It is clear from the curl 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,

(A.164) |

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,

(A.165) |

Thus, a conservative field is also an irrotational one.