Stokes theorem concept question

Question

I've got a conceptual question about Stokes theorem. So the way I understood it Stokes theorem is used to calculate the counterclockwise circulation through a smooth oriented surface. However one of the questions at the end of the asks to find the Flux through a given surface using Stokes theorem. I wasn't sure how this could be done, in my understanding I thought that is what Gauss's Divergence Theorem is for, but I could be mistaken.

Answer 1

What Stokes' Theorem gives you is the relation between the surface integral of the curl of a vector field over a smooth oriented surface S, to the line integral of the vector field over its boundary C:

$$\int_C \vec F \cdot \vec {ds} = \iint_S (\nabla \times \vec F) \cdot \vec {dS}$$

The key here is to realize that the question they're asking you, to calculate the Flux integral over the surface S using Stokes theorem can be formulated as follows:

Let $ \vec G= \nabla \times \vec F$ then $$\iint_S \vec G \cdot \vec {dS} = \iint_S \nabla \times \vec F \cdot \vec {dS} = \int_C \vec F \cdot \vec {ds}$$ and from here what you should be able to is to find $F$ such that $G=\nabla \times \vec F$ and calculate the line integral. (Note that this can be done if and only if $\nabla \cdot \vec G=0$ )

Hope that helps!