alternate characterization of curl
Let be a smooth vector field on (an open subset of) .
We show that defined using the coordinate-free definition given on the parent entry (http://planetmath.org/curl) is the same as the curl defined by in Cartesian coordinates.
The case for spherical surfaces
Combining these facts, we obtain
Notice that depends only on the derivative of at .
We want to evaluate the last integral in Cartesian coordinates. Let be an orthonormal basis of oriented positively, and let be the matrix of the derivative in this basis. Then the th coordinate of with respect to the same basis is
The th coordinate of the integrand is
In the summation above, if a summmand has , then the integral of that summand over the sphere is zero, by symmetry. This means that in the summation the index may be set to , and thus
Now there is a formula for the evaluation of integrals of polynomials over , in terms of the gamma function; in our case () the formula reads:
But this is just .
The case for arbitrary surfaces
Although we have only computed
only for spheres , this formula holds for arbitrary closed surfaces that shrink nicely to . It is hardly obvious, especially since our computation before depended on the symmetry of the sphere extensively.
To show the general result, consider the triple scalar product . This is a linear functional in the vector , so there exists a unique vector function such that for all . We can find the components of this by evaluating the functional at :
The reason for considering such expressions is that, putting , we have
So we have converted the original integral into an ordinary surface integral. And this surface integral can be changed into a volume integral, by using the divergence theorem:
where is the volume whose boundary is . Hence
Definition in terms of differential forms
We mention, in passing, a computational, yet coordinate-free, alternative to the definition of the curl, using differential forms. If is a 1-form on such that , then the curl of is defined as the vector function such that
In Cartesian coordinates, we have
|Title||alternate characterization of curl|
|Date of creation||2013-03-22 15:29:10|
|Last modified on||2013-03-22 15:29:10|
|Last modified by||stevecheng (10074)|