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Integrationg over exp with two variables

  1. Nov 11, 2013 #1
    1. The problem statement, all variables and given/known data
    f(x,y) = exp(-x^2 +xy -y^2)

    transform with
    x =(1/sqrt(2)) *(u – v), y = (1/sqrt(2))* (u + v) .


    2. Relevant equations

    Jacobian

    3. The attempt at a solution

    Jacobian = 1

    f(u,v) = exp(-(u^2)/2 -(3v^2/2)

    double integral f(u,v) du dv

    the bounds would be x > 0 => ( u-v) >0 => u > v
    and x < ∞ => u < ∞

    v > 0 to v < ∞

    I am lost on what to do next. If anyone can be as kind as to help, I would greatly appreciate it!
     
  2. jcsd
  3. Nov 12, 2013 #2

    tiny-tim

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    hi there cutesteph! :smile:

    (try using the X2 button just above the Reply box :wink:)

    what are your limits for x and y ? :confused:

    i'll assume they're both from 0 to ∞

    draw the region (in x,y), and mark a grid of lines of equal u and v

    u goes from 0 to ∞

    for each value of u, where does v go from and to? :wink:
     
  4. Nov 12, 2013 #3
    So the limits v from 0 to infinity and u from -v to v.

    ∫0 to∞ exp(-u2/2)∫-u to u exp(-3v2/2) dv du
     
  5. Nov 12, 2013 #4

    tiny-tim

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    isn't it the other way round?
     
  6. Nov 12, 2013 #5

    Ray Vickson

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    You never actually answered the question about the limits on x and y, and without your answer I cannot possibly tell what are the limits on u and v. However, you can determine the latter for yourself by noting that
    [tex] u = \frac{x+y}{\sqrt{2}}, \; v = \frac{y-x}{\sqrt{2}} [/tex]
    If you know the ranges of x and y you can figure out the ranges on u and v.
     
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