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Integration by change of variable

  1. Jun 27, 2008 #1
    Hello:

    In Problem Solution part below, I am not sure of Step 2 and am having problems with Step 3.
    Thanks in advance


    Problem Statement:

    Let D = {(x1,x2)| x1,x2>0, 1<= x1[tex]^{2}[/tex] - x2[tex]^{2}[/tex] <=9, 2 <= x1x2 <= 4.
    Use hyperbolic coordinates g(u,v) = {u[tex]^{2}[/tex] - v[tex]^{2}[/tex], uv} to show that

    integ on D of (x1[tex]^{2}[/tex] + x2[tex]^{2}[/tex]) dx1dx2 = 8 ---- equation 1.


    Problem Solution:


    Step1: To show that Jacobian of g is non-zero
    This was OK. Det is 2u[tex]^{2}[/tex] + 2v[tex]^{2}[/tex]. it is also given that u,v >0. Therefore det will always be >0


    Step2: To convert x1,x2 into u, v

    Substituting (u[tex]^{2}[/tex] - v[tex]^{2}[/tex])[tex]^{2}[/tex] into x1[tex]^{2}[/tex] and (uv)[tex]^{2}[/tex] into x2[tex]^{2}[/tex] equation 1 changes to after manipulation to ( I also did multiply by the Jacobian)

    integ on R of (2u[tex]^{6}[/tex] + 2v[tex]^{6}[/tex]

    Step3: To determine limits of u,v.
    This is where I am having problems with

    How do I translate 1<= x1^2 - x2^2 <=9 and 2 <= x1x2 <= 4 into u, v. Everything I do doesn't give me the ans in the question.



    Thanks

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



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 28, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You are taking x1= u2- v2 and x2= uv?

    It is far simpler to let u= x12- x22 and v= x1x2.
     
  4. Jun 29, 2008 #3
    Excellent - thank you.
    It worked.
     
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