# Gauss' Theorem

• May 6th 2010, 01:19 PM
davesface
Gauss' Theorem
Let $S_r$ denote the sphere of radius r with center at the origin, oriented with outward normal. Suppose F is of class $C^1$ on all of $\mathbb{R}^3$ and is such that $\oint\oint F \cdot dS=ar+b$ for some fixed constants a and b.

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

b) Suppose, in the situation just described, that $F=\nabla \times G$ for some vector field G of class $C^1$. 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?
• May 6th 2010, 04:51 PM
zzzoak
$\int \int \int_D \nabla \cdot F dV \: = \: \oint \oint F \cdot dS = \int \int_{r=5} F \cdot dS \: + \: \int \int_{r=7} F \cdot dS =$
$= \: - (5a+b) + 7a + b = 2a$

As
$
\nabla \cdot \nabla \times G \: = \:0
$

$
ar+b=0
$

$
a=0
$

$
b=0.
$
• May 7th 2010, 04:38 PM
zzzoak
Sorry, I think that
a=0
b=0
is in $D'=(x^2+y^2+z^2<=r^2)$
but in D
2a=0 from first equation I wrote
and we get a=0 and b=const.