When is the vector surface integral through the surface a sphere 0 by symmetry?

Question

When is the flux through a sphere 0 by symmetry? I calculated the flux through a sphere using the vector field $ (x^2,0,0)$ through some arbitrary sphere to be 0. Is this a special case of a larger situation?

Answer 1

Note that as a consequence of Gauss's divergence theorem

$$\iiint_V \nabla\cdot \vec{F} \, dV= \iint_S \vec{F}\cdot d\vec{S}$$

in this case

$$\nabla\cdot(x^2,0,0)=2x$$

thus the flux integral of $(x^2,0,0)$ through a surface of a symmetric domain with respect to y-z plane is always zero.