Results 1 to 3 of 3

Math Help - Gauss' Theorem

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    88

    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    Posts
    280
    \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
    <br />
 \nabla \cdot \nabla \times G \: = \:0 <br />
    <br />
ar+b=0<br />
    <br />
a=0<br />
    <br />
b=0.<br />
    Last edited by zzzoak; May 6th 2010 at 04:49 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Mar 2010
    Posts
    280
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Gauss's theorem, gradient
    Posted in the Calculus Forum
    Replies: 0
    Last Post: April 1st 2011, 10:43 AM
  2. A theorem of Gauss
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: June 7th 2010, 04:01 PM
  3. Gauss Divergence Theorem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 13th 2010, 05:04 AM
  4. Gauss Divergence Theorem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 11th 2010, 06:29 AM
  5. Variation of Gauss' Theorem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 21st 2007, 11:36 AM

Search Tags


/mathhelpforum @mathhelpforum