Let

.

Calculate the flux of the curl of

through the surface of the semi-sphere

,

, oriented positively. Use Stokes theorem. Then use another way that does not involve Stokes' theorem to solve the problem.

My attempt :

.

.

.

Hence

. How is that possible? The field is not conservative and the surface is closed, hence it is impossible that the net flux passing through the semi-sphere give

.

I want to tackle it by calculating

but with no success. I parametrize

:

. Thus

.

Now I don't know what to do, is it a line integral of a scalar field or vector field? Seems more like a vector field to me, let's try.

. What's wrong?!

Using the divergence theorem I got

. However I do not know where to check if I didn't made a mistake of sign, since they said the surface to be oriented positively. The divergence is worth

so it can be positive or negative.

I'm all confused. What's the difference between Stokes and Gauss' theorems? They can both calculate the flux through a surface, hence they're equivalent?!