
The Divergence Theorem
I'm not really sure why this is...
Let F(x,y,z) be a nonzero vector field in 3space whose component functions have continuous first partial derivatives, and assume that
div F = 0 everywhere. If σ is any sphere in 3space, explain why there are infinitely many points on σ at which F is tangent to the sphere.

If div F = 0 then the field flux who enters the sphere must be the same as the field flux which leaves the sphere.
as the field is continuous and non zero, there will be in the surface, between 2 points (1 with negative and other with positive flux), a point with zero flux which is when the field is tangent.
also as you can find an infinity number of paths between those 2 points which do not cross each others there will be an infinty of points with flux zero. (this is seen also by the sphere symmetry)