Gauss' Theorem

**davesface** · May 6th 2010, 12:19 PM

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?

**zzzoak** · May 6th 2010, 03:51 PM

$\int \int \int_D \nabla \cdot F dV \: = \: \oint \oint F \cdot dS = \int \int_{r=5} F \cdot dS \: + \: \int \int_{r=7} F \cdot dS =$
$= \: - (5a+b) + 7a + b = 2a$

As
$ \nabla \cdot \nabla \times G \: = \:0 $
$ ar+b=0 $
$ a=0 $
$ b=0. $

**zzzoak** · May 7th 2010, 03:38 PM

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.

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