Let $\displaystyle S_r$ denote the sphere of radius r with center at the origin, oriented with outward normal. Suppose F is of class $\displaystyle C^1$ on all of $\displaystyle \mathbb{R}^3$ and is such that $\displaystyle \oint\oint F \cdot dS=ar+b$ for some fixed constants a and b.

a) Compute $\displaystyle \int \int \int_D \nabla \cdot F dV $, where $\displaystyle D=\{(x,y,z)|25\leq x^2+y^2+z^2\leq 49\}$. Your answer should be in terms of a and b.

b) Suppose, in the situation just described, that $\displaystyle F=\nabla \times G$ for some vector field G of class $\displaystyle C^1$. What conditions does this place on the constants a and b?

So I see that the integral is going to be over the region between the spheres of radius 5 and radius 7, but how should I proceed given that F is not explicitly defined?