1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook