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Homework Help: Jacobi Matrix and Multiple Intgrals

  1. Mar 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Let D be the set of points (x,y) in R^2 for which 0 is ≤ x ≤ 1 and 0 ≤ y ≤ 1. Find a function g: R^2 --> R for which:

    ∫_0^1 ∫_0^1 h(x,y)dxdy = ∫_0^1∫_0^1 h(y^5, x^3) * g(x,y)dxdy

    is true for all functions h: D--> R integrable over D

    In the question before this I was asked to find a jacobi matrix and determinant for f(x,y) = (y^5,x^3)

    I found that the determinant is -15y^4x^3
    2. Relevant equations

    3. The attempt at a solution

    ∬_D f(x,y)dxdy= ∬_S g(v^5,u^3) * -15y^4x^2

    Is this my final result? If not, can I get some help on how to go on?
  2. jcsd
  3. Mar 29, 2012 #2


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    Science Advisor
    Homework Helper
    Gold Member

    Your answer shouldn't have 4 variables and needs the integration variables.

    Your problem certainly suggests the change of variables ##x=u^5,\ y=v^3##. You need to check that maps the square to the square. Then use your change of variables theorem to express your integral in terms of ##u## and ##v##. Your result should have just ##u## and ##v## variables in it. Of course, they are dummy variables in the final result.
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