Results 1 to 3 of 3
Like Tree1Thanks
  • 1 Post By ebaines

Math Help - Checking the divergence theorem on a sphere

  1. #1
    Junior Member BERMES39's Avatar
    Joined
    Aug 2011
    Posts
    41

    Checking the divergence theorem on a sphere

    I have not been able to solve this problem. Can someone please help me find the error?
    Attached Thumbnails Attached Thumbnails Checking the divergence theorem on a sphere-ch1.3-1.39.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor ebaines's Avatar
    Joined
    Jun 2008
    From
    Illinois
    Posts
    1,064
    Thanks
    296

    Re: Checking the divergence theorem on a sphere

    I think your problem is that  \bigtriangledown \cdot r^2 \hat r is not 2r, but rather is 4r. In spherical coordinates you have:

     \bigtriangledown \cdot \vec v = \frac 1 {r^2} \frac {\partial (r^2 v_r)}{\partial r} + \frac 1 {r \sin \theta} \frac {\partial (v_\theta \sin \theta)}{\partial \theta} + \frac 1 {r \sin \theta} \frac {\partial v_\phi}{\partial \phi}

    For the case of \vec v=r^2 \hat r this becomes:

     \bigtriangledown \cdot \vec v = \frac 1 {r^2} \frac {\partial (r^4)}{\partial r} = 4r

    Use this to determine  \int _{Vol} ( \bigtriangledown \cdot \vec v ) dV and it turns out to be  4 \pi R^4.
    Last edited by ebaines; January 14th 2014 at 10:31 AM.
    Thanks from BERMES39
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member BERMES39's Avatar
    Joined
    Aug 2011
    Posts
    41

    Re: Checking the divergence theorem on a sphere

    Thank you very much!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Stoke's theorem and the Divergence theorem.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 2nd 2012, 04:16 PM
  2. Green's theorem question needs checking
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 19th 2010, 07:42 AM
  3. Replies: 3
    Last Post: May 14th 2010, 10:04 PM
  4. Replies: 2
    Last Post: April 3rd 2010, 04:41 PM
  5. Mass of a sphere, checking my result
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 20th 2009, 05:52 PM

Search Tags


/mathhelpforum @mathhelpforum