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Thread: Gauss' Theorem

  1. #1
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    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?
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  2. #2
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    $\displaystyle \int \int \int_D \nabla \cdot F dV \: = \: \oint \oint F \cdot dS = \int \int_{r=5} F \cdot dS \: + \: \int \int_{r=7} F \cdot dS =$
    $\displaystyle = \: - (5a+b) + 7a + b = 2a$

    As
    $\displaystyle
    \nabla \cdot \nabla \times G \: = \:0
    $
    $\displaystyle
    ar+b=0
    $
    $\displaystyle
    a=0
    $
    $\displaystyle
    b=0.
    $
    Last edited by zzzoak; May 6th 2010 at 04:49 PM.
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  3. #3
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    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.
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