
Gauss' Theorem
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?

$\displaystyle \int \int \int_D \nabla \cdot F dV \: = \: \oint \oint F \cdot dS = \int \int_{r=5} F \cdot dS \: + \: \int \int_{r=7} F \cdot dS =$
$\displaystyle = \:  (5a+b) + 7a + b = 2a$
As
$\displaystyle
\nabla \cdot \nabla \times G \: = \:0
$
$\displaystyle
ar+b=0
$
$\displaystyle
a=0
$
$\displaystyle
b=0.
$

Sorry, I think that
a=0
b=0
is in $\displaystyle D'=(x^2+y^2+z^2<=r^2)$
but in D
2a=0 from first equation I wrote
and we get a=0 and b=const.