How to evaluate $\int_{S^2} \sin(\theta) d\theta \wedge d\phi$?

Question

I'm kind of stuck here. I try to apply Stokes Theorem, so that $\int_{S^2} \sin(\theta) d\theta \wedge d\phi = \int_{D^2} d(\sin(\theta) d\theta \wedge d\phi) = 0 $ which is not the answer?

Answer 1

You mean $D^3$, of course. Try taking $\omega = r^3\sin(\theta)d\theta\wedge d\phi$ instead. The scale-invariant area form you wrote down is, of course, closed.

In cartesian coordinates, your form is $$\frac{x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy}{(x^2+y^2+z^2)^{3/2}}.$$ You could just eliminate the denominator ... Note that Stokes's Theorem doesn't apply until you eliminate the denominator.