I am working on a few surface integrals in preparation for an exam and one question specifically states to use Stokes' Theorem to solve, however, rather than giving a vector field, we are given a differential two-form. I know the generalized Stokes'-Cartan Theorem, but this problem specifically states to use just Stokes' Theorem. Is this possible? If so, how?
The question I'm specifically working on is to evaluate: $$ \int_{S_2}z\ dy\ dz $$ where $S_2$ is the intersection between $S$ the unit sphere and $z\le1$. I am thinking that the boundary will be the part of the sphere below $z=1$ plus the circle at $z=1$ with radius $r=\sqrt3$. I'm not sure if this is right, however, as in the vector calculus version of Stokes' Theorem, it relates a surface integral to a line integral. Any help is appreciated!