Gauss' Theorem

• May 6th 2010, 12:19 PM
davesface
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?
• May 6th 2010, 03:51 PM
zzzoak
$\displaystyle \int \int \int_D \nabla \cdot F dV \: = \: \oint \oint F \cdot dS = \int \int_{r=5} F \cdot dS \: + \: \int \int_{r=7} F \cdot dS =$
$\displaystyle = \: - (5a+b) + 7a + b = 2a$

As
$\displaystyle \nabla \cdot \nabla \times G \: = \:0$
$\displaystyle ar+b=0$
$\displaystyle a=0$
$\displaystyle b=0.$
• May 7th 2010, 03:38 PM
zzzoak
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.