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: 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

    User Avatar
    Science Advisor

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