I am asked to compute an integral using (I guess) Stokes thereom. Here is the statement : let $S\subset \mathbb{R}^3$ given by $S=\{x,y,z : x^2+y^2+z^2=1\}$ and $X:\mathbb{R}^3\to\mathbb{R}^3$ a vector field given by $X(x,y,z)=(x,y,z)$. Compute $\iint_S \text{rot}(X)\cdot \nu \,d\sigma$, where $\nu$ is the interior normal, meaning $\nu$ points to the origin.
What I did is that I splitted the integral in two pieces (upper half sphere and lower half sphere), and then by using Stokes theorem, I have the sum two integrals that are opposite of each other, since one time the boundary is travelled clockwise, and the other time it is travelled counterclockwise. Is that true ? I feel that this can work given any vectorfield $X$, so I am not sure.