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Green's Theorem

  1. Nov 3, 2005 #1
    Prove the identity [tex]\int_{\partial D}\phi \nabla \phi \cdot \n \ds = \int \int_{D} (\phi \nabla^2 \phi + \nabla \phi \cdot \nabla \phi) \dA [/tex]
    Can I just let [tex]\phi[/tex] be equal to P + Q, substitute into the left side, and try to derive the right side? This is a weird looking identity by the way.
    Last edited: Nov 3, 2005
  2. jcsd
  3. Nov 4, 2005 #2
    Its actually quite easy, without any vector calculus needed.
    (f*gx)x= fx*gx + f*gxx

    if you sum these up for y and z derivatives, you will get

    Div[f*del(g)] = del(f) dot del(g) + f*del^2(g)

    integrate both sides and use divergence theorem.

    I apologize for having to write div for divergence, dot for dot product and del for the del operator, but i dont know how to do those fancy fonts.
  4. Nov 4, 2005 #3
    So basically let F be equal to [tex] \phi \nabla \phi [/tex], then [tex] \nabla \cdot F [/tex] is equal to [tex](\phi \nabla^2 \phi + \nabla \phi \cdot \nabla \phi) [/tex]. What do you know it's the divergence theorem!
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