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Closed curve line integral of gradient using Green's Theorem

  1. Apr 22, 2010 #1
    Apostol page 386, problem 5
    1. The problem statement, all variables and given/known data
    Given [tex]f,g[/tex] continuously differentiable on open connected [tex]S[/tex] in the plane, show
    [tex]\oint_C{f\nabla g\cdot d\alpha}=-\oint_C{g\nabla f\cdot d\alpha}[/tex]
    for any piecewise Jordan curve [tex]C[/tex].

    2. Relevant equations
    1. Green's Theorem
    2. [tex]\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}[/tex] for [tex]\nabla f,\nabla g[/tex]

    3. The attempt at a solution
    I need some general direction on this one...
    Last edited: Apr 22, 2010
  2. jcsd
  3. Apr 22, 2010 #2


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    Use that grad(fg)=f*grad(g)+g*grad(f), maybe?
  4. Apr 24, 2010 #3
    that makes it about a 2 second proof then
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