Flow of the vector field with surface integral

Question

We are asked to calculate with the help of a suitable surface integral, the flow of the vector field F=(z,y,x) through the sphere with center (0,0,0) and radius 1, from the center of the sphere and outwards.

I am really aware what exactly to do to solve it, but should I implement Stokes theorem, or it can be solved only through parametrization? Thank you very much in advance.

Answer 1

As noted in the comments, the outward unit normal field on the unit sphere $S$ is $n = (x, y, z)$, so $F \cdot n = 2xz + y^2$. We need to integrate this function over $S$, a task that can be done by symmetry and geometry:

• The integral over $S$ of a function that is odd in any variable is $0$.
• By symmetry, $$ \iint_{S} y^{2}\, dS = \iint_{S} x^{2}\, dS = \iint_{S} z^{2}\, dS. $$ But on the unit sphere, $$ \iint_{S} (x^{2} + y^{2} + z^{2})\, dS = \iint_{S} 1\, dS, $$ which is the area of $S$.