Integrating 2-Forms on the Unit Sphere

[raw]

Homework Statement

I want to integrate the 2-form defined on R^3\{0,0,0} over the unit sphere.
(x/r^3)dy wedge dz+(y/r^3)dz wedge dx+(z/r^3)dx wedge dy

Homework Equations

r=[tex]\sqrt{x^2+y^2+z^2}[/tex]

The Attempt at a Solution

I'm thinking this is like a surface integral but I'm not really sure how to go about actually doing the calculation.

 

[hunt_mat]

First thing to do is convert the 2-form into polar form.

 

[Hurkyl]

As hunt_mat said, you could just do the direct thing: parametrize the surface and integrate. He suggests polar coordinates; I'm not sure if he really meant cylindrical coordinates or spherical coordinates, though. It might be worth considering plain ordinary rectangular coordinates; the three summands are pretty much already set up as ordinary double integrals.


Normally you'd consider the generalized Stokes' theorem to integrate this. (Green's theorem?) But, alas, the origin is a problem.

So what if you put a tiny sphere around the origin, and used Stokes' theorem on the region between them? Or alternatively a huge sphere. If you can say something useful about the behavior of the integral on very small or very large spheres, this approach could work.

 

[hunt_mat]

I meant spherical co-ordinates, as Hurkyl said the origin is a problem so, take a small sphere of radius epsilon and exclude that from the domain of integration and then once you have done the integration, take the limit as epsilon tends to zero.

 

[Hurkyl]

Oh, and by the way, can't you simplify the integrand?