Final answer:
The statement that if the surface integral of a vector field over a sphere is zero, then the divergence of that field is zero at all points inside the sphere, is false. This is because the zero net flux through the sphere does not necessitate zero divergence within; it depends on the charge distribution inside the sphere.
Step-by-step explanation:
The statement 'If s is the unit sphere oriented outward and F is a vector field satisfying ∫ₙs F . dA = 0, then divF = 0 at all points inside s' is False. The integral given is the surface integral of the vector field F over the sphere s, and the fact that it equals zero only implies that the net flux of F through the surface is zero.
This, however, does not necessarily mean that the divergence of F inside the sphere is zero. According to Gauss's Law, the surface integral of a vector field over a closed surface is equal to the volume integral of the divergence of the field inside the surface, which is related to the charge enclosed inside the surface.
For divF to be zero everywhere inside s, it must be the case that F has zero divergence everywhere within that region, which could occur if there are no charges or if charges are distributed such that their contributions cancel out exactly. Without additional information on the distribution of sources within the sphere, one cannot conclude that divF is zero everywhere inside based solely on the net flux being zero.