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Homework Help: Evaluating double integral - jacobian help

  1. May 9, 2010 #1

    bon

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    1. The problem statement, all variables and given/known data

    trying to evaluate the double integral from 0 to infinity and 0 to infinity of [(x^2 + y^2)/1 + (x^2-y^2)^2]e^-2xy dxdy

    using the coordinate transformation u=x^2-y^2 and v=2xy



    2. Relevant equations



    3. The attempt at a solution

    so i calculated the jacobian which looks nice 4(x^2+y^2)..can see some canceling there

    just can't see what the new limits will be...

    thanks for any help
     
  2. jcsd
  3. May 9, 2010 #2

    tiny-tim

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    Hi bon! :smile:

    (have an infinity: ∞ and try using the X2 tag just above the Reply box :wink:)

    When you're trying to find new 2D or 3D limits, just draw the region, and then mark it with the "contour lines" of the new variables.

    In this case, the region is the whole first quadrant …

    now draw some typical curves for u = constant and for v = constant …

    check that (u,v) is single-valued, and just read off the diagram what the lowest and highest "contour lines" are. :wink:
     
  4. May 9, 2010 #3

    bon

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    Re: Jacobian

    Ok thanks so i see that v goes from 0 to infinity...i just can't see u at the moment..

    thanks
     
  5. May 9, 2010 #4

    tiny-tim

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    u = 0 is the straight diagonal line …

    what are the other curves for u = constant in the first quadrant? :wink:
     
  6. May 9, 2010 #5

    bon

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    Re: Jacobian

    ahh hyperbolae..so would it be -infinity to + infinity?

    thanks
     
  7. May 9, 2010 #6

    bon

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    Re: Jacobian

    is this right?
     
  8. May 9, 2010 #7

    tiny-tim

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    let's see …

    each lower "hyperbola" is a quarter of a hyperbola, starting at the x-axis and finishing "at infinity", close to the diagonal …

    so it goes from xy = 0 to xy = ∞ (and the same for the upper "hyperbolas").

    So yes, x2 - y2 goes from -∞ to ∞, and for each value of x2 - y2, xy goes from 0 to ∞. :smile:
     
  9. May 9, 2010 #8

    bon

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    Re: Jacobian

    great thanks
     
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