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Rewrite Indefinite Integral in Terms of Elliptic Coordinates

  1. Sep 9, 2013 #1
    Problem:

    Rewrite the indefinite integral ## \iint\limits_R\, (x+y) dx \ dy ## in terms of elliptic coordinates ##(u,v)##, where ## x=acosh(u)cos(v) ## and ## y=asinh(u)sin(v) ##.


    Attempt at a Solution:

    So would it be something like,

    ## \iint\limits_R\, (x+y) dx \ dy = \iint\limits_R\, [acosh(u)cos(v)+asinh(u)sin(v)] dx \ dy ##. I need to calculate ##dx## and ##dy## in terms of ##u## and ##v##, then substitute them in the integral, correct?

    The only thing that comes to mind is the multivariable chain rule: ## \frac{dx}{dt} = \frac{\partial x}{\partial u} \frac{du}{dt}+\frac{\partial y}{\partial v} \frac{dv}{dt} ## or ## \frac{dy}{dt} = \frac{\partial y}{\partial u} \frac{du}{dt}+\frac{\partial y}{\partial v} \frac{dv}{dt} ##

    Any help is appreciated. Thanks in advance.
     
  2. jcsd
  3. Sep 9, 2013 #2
    I should note that the task is to merely rewrite the integral, not to evaluate it.
     
  4. Sep 9, 2013 #3

    vela

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    Think Jacobian.
     
  5. Sep 9, 2013 #4
    I'm sorry, can you be more specific?
     
  6. Sep 9, 2013 #5

    vela

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  7. Sep 9, 2013 #6
    I think I got it, hold on.
     
  8. Sep 9, 2013 #7
    So I simply use ## \iint\limits_R\ f(x,y) dx \ dy = \iint\limits_{R^*}f(x(u,v),y(u,v)) det(J) du \ dv ##?
     
  9. Sep 9, 2013 #8

    vela

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    Yup.
     
  10. Sep 9, 2013 #9
    Okay thanks! I'm gonna LaTeX my work so you can see.
     
  11. Sep 9, 2013 #10
    For the Jacobian I got ## J = (a^2sinh^2(u)cos^2(v)-a^2cosh(u)sinh(u)cos(v)sin(v))du \ dv ##. Is that correct?
     
  12. Sep 9, 2013 #11

    vela

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    I don't get that.
     
  13. Sep 9, 2013 #12
    Darn. I'll try agian.
     
  14. Sep 9, 2013 #13
    I think I messed up a negative. Now I'm getting ## J= (asinh(u)cos(v))^2+(acosh(u)sin(v))^2 ##
     
  15. Sep 9, 2013 #14

    vela

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    I'm not at my computer right now, but it looks like what I got.
     
  16. Sep 9, 2013 #15
    Awesome! Thank you so much!
     
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