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Math Help - Jacobian transformation

  1. #1
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    Jacobian transformation

    Suppose that the double integral over D of f(x,y)dA-1 where D is the disk x^2+y^2_<9 Now suppose E is the disk x^2+y^2_<36 and g(x,y)=4f(x/2,y,2) What is the value of the double integral over E of g(x,y) dA?
    Please help
    thank you
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  2. #2
    Super Member
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    Aug 2008
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    I'd like to make an attempt at this even though I don't think it's what you want:

    Consider:

    \mathop\int\int\limits_{\hspace{-22pt}x^2+y^2=9} f(x,y)dA

    I'll use the following change of variables:

    u=\frac{x}{2},\quad v=\frac{y}{2}

    x=2u,\quad v=2u

    Then:

    J=\left|\begin{array}{cc}2 & 0 \\ 0 & 2\end{array}\right|=4

    Substituting the values of u and v into the circle in terms of x and y I get the new domain in terms of u and v:

    u^2+v^2=9/4

    Thus:

    \mathop\int\int\limits_{\hspace{-22pt}x^2+y^2=9} f(x,y)dA=4\mathop\int\int\limits_{\hspace{-22pt}u^2+v^2=9/4} f(u,v)dA

    (Would be nice to have a real double and triple integral sign in here . . . )
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