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Homework Help: Prove the change of variables formula for double integrals

  1. Apr 12, 2010 #1
    1. The problem statement, all variables and given/known data
    Use Green's Theorem to prove for the case f(x,y) = 1

    [tex]\int\int_R dxdy = \int\int_S |\partial(x,y)/\partial(u,v)|dudv[/tex]

    EDIT: R is the region in the xy-plane that corresponds to the region S in the uv-plane under the transformation given by x = g(u,v), y = h(u,v), and the expression on the right is the Jacobian.

    2. Relevant equations
    [tex]A = \oint _C xdy = -\oint _C ydx = 1/2\oint _C xdy - ydx[/tex]


    3. The attempt at a solution
    My textbook says that the left side of the equation I'm trying to prove is A(R), so I can apply the first part of the equation in 2., but I don't understand what it's talking about because the equation in 2. deals with line integrals, and I'm trying to prove an equation with double integrals.
     
    Last edited: Apr 12, 2010
  2. jcsd
  3. Apr 12, 2010 #2

    Mark44

    Staff: Mentor

    What is R? S?
    How are u and v related to x and y?
    Isn't the expression in absolute values on the right side the determinant of the Jacobian?
     
  4. Apr 12, 2010 #3
    R is the region in the xy-plane that corresponds to the region S in the uv-plane under the transformation given by x = g(u,v), y = h(u,v), and, yes, that is the Jacobian.
     
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