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Math Help - Jacobian Integration Problem

  1. #1
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    Jacobian Integration Problem

    \int \int e^(-r(x-2y)^2) dxdy

    x =2u+v
    y=u-v

    The Jacobian is -3
    I've substituted for x and y in the integral to get \int \int -3e^r(3v)^2 dudv

    where r is a positive constant

    But my problem lies with the limits, I have y is greater than or equal to 0 and x is less than or equal to 0, How do I change these for the integral and proceed as a result?

    Thanks
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  2. #2
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    Quote Originally Posted by llamagoogle View Post
    \int \int e^(-r(x-2y)^2) dxdy

    x =2u+v
    y=u-v

    The Jacobian is -3
    I've substituted for x and y in the integral to get \int \int -3e^r(3v)^2 dudv

    where r is a positive constant

    But my problem lies with the limits, I have y is greater than or equal to 0 and x is less than or equal to 0, How do I change these for the integral and proceed as a result?

    Thanks

    What you wrote in the 1st integral doesn't show clearly. Anyway:

    \left\{\begin{array}{c}x=2u+v\\y=\;\;u-v\end{array}\right. \Longrightarrow x+y=3u\Longrightarrow u=\frac{x+y}{3}\Longrightarrow v=\frac{x-2y}{3} , so -\infty < u<\infty\,,\,\,-\infty<v\leq 0

    Tonio
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  3. #3
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    Hello,

    Hmmm the Jacobian has to be the absolute value of the determinant, so it would be 3.

    now for the boundaries :

    we have x=2u+v\leq 0 and y=u-v\geq 0

    from this, it follows that \boxed{v\leq u\leq -\tfrac v2}

    and for the boundaries of v : v\leq -\tfrac v2 \Rightarrow \boxed{v\leq 0}
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