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Gradient and Divergent Identities

  1. Jul 15, 2012 #1
    1. The problem statement, all variables and given/known data

    I need to show that ##\displaystyle\int_\Omega (\nabla G)w dxdy=-\int_\Omega (\nabla w) G dxdy+\int_\Gamma \hat{n} w G ds## given

    ##\displaystyle \int_\Omega \nabla F dxdy=\oint_\Gamma \hat{n} F ds## where ##\Omega## and ##\Gamma## are the domain and boundary respectively. F,G and w are scalar functions...any ideas?

    I attempted to expand the LHS but I didnt feel it was leading me anywhere...

    2. Relevant equations
    3. The attempt at a solution

    ##\displaystyle \int_\Omega (\hat{e_x}\frac{\partial G}{\partial x}+\hat{e_y}\frac{\partial G}{\partial y})w dx dy##....

    NOTE: I have posted this query on MHF 3 days ago and nobody has answered. Here is the link just in case somebody has replied. thanks http://mathhelpforum.com/calculus/200911-gradient-divergent-identities.html
     
  2. jcsd
  3. Jul 15, 2012 #2

    HallsofIvy

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    Integration by parts, taking u= w, [itex]dv= \nabla G dx dy[/itex].
     
  4. Jul 15, 2012 #3
    If we let ##u=w## then ##du=dw=\nabla w##???

    ##\displaystyle dv=\nabla G dxdy## then

    ##\displaystyle v=\int_\Omega \nabla G dxdy=\int_\Gamma (\hat{n_x} \hat{e_x}+ \hat{n_y} \hat{e_y})Gds##

    Thus

    ##\displaystyle ∫_Ω(∇G)wdxdy= \int_\Gamma (\hat{n_x} \hat{e_x}+ \hat{n_y} \hat{e_y})G w ds- \int \int_\Gamma (\hat{n_x} \hat{e_x}+ \hat{n_y} \hat{e_y})G \nabla w ds##

    Clearly I have gone wrong somewhere....?
     
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