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Thread: 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 $\displaystyle x^2+y^2_<9$ Now suppose E is the disk $\displaystyle x^2+y^2_<36$ and $\displaystyle g(x,y)=4f(x/2,y,2)$ 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
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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:

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

    I'll use the following change of variables:

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

    $\displaystyle x=2u,\quad v=2u$

    Then:

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

    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:

    $\displaystyle u^2+v^2=9/4$

    Thus:

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

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