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Homework Help: Help solving Green's identities question.

  1. Jul 30, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose [itex]u[/itex] is harmonic ([itex]\nabla^2 u = 0 [/itex]) and [itex]v=0 \;\hbox{ on } \;\Gamma [/itex] where [itex]\Gamma[/itex] is the boundary of a simple or multiply connected region and [itex] \Omega[/itex] is the region bounded by [itex]\Gamma[/itex].

    Using Green's identities, show:

    [tex] \int \int_{\Omega} \nabla u \cdot \nabla v \; dx dy = 0 [/tex]




    2. Relevant equations

    Green's identities:

    [tex] \int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds [/tex]

    [tex] \int \int_{\Omega} \;(u\nabla^2 v \;-\; v\nabla^2 u )\; dx dy \;=\; \int_{\Gamma} \;(u\frac{\partial v}{\partial n} - v\frac{\partial u}{\partial n}) \;ds [/tex]

    [tex] \frac{\partial u}{\partial n} = \nabla u \; \cdot \; \widehat{n}[/tex]

    3. The attempt at a solution

    I use

    [tex] \int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds [/tex]

    If [itex]v=0 \hbox { on } \Gamma \;\;\Rightarrow\;\; v \hbox { is a constant = 0 } \;\;\Rightarrow\;\; v= 0 \;\hbox{ on } \;\Omega[/itex].

    [tex] \int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \int \int_{\Omega}\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds = 0[/tex]

    Is this the right way?
     
    Last edited: Jul 31, 2010
  2. jcsd
  3. Jul 31, 2010 #2
    I think I have the answer, I edit the original post already. Can anyone verify my work?
     
    Last edited: Jul 31, 2010
  4. Aug 1, 2010 #3
    Anyone please?
     
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