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Multivariable Transformations

  1. Nov 28, 2008 #1
    I was reading a statistics book, and part of the problem reduces to the calculus problem of doing the following:

    1) Let u=x/y, v=y, with domain 0<x<y<1, how to find the ranges of u and v after the transformation?


    2) Let u=x/(x+y), v=x+y with domain x>1, y>1, what values can u and v take on?


    Is there a systematic way to do these?

    Thank you for any help!
     
  2. jcsd
  3. Nov 30, 2008 #2
    How can I determine step-by-step the corresponding region in the uv-plane after the transformation?
     
  4. Dec 3, 2008 #3
    Please help...
     
  5. Dec 3, 2008 #4

    HallsofIvy

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    The range of v should be obvious. Is it possible for u to be negative? Is it possible for u to be 0? So what is a lower bound for u? Since y can be as close to 0 as you please is there an upper bound on u?

    Another, more general, way to do this is to look at the boundary lines. If x= 1, then u= 1/y and v= y so u= 1/v. Graph u= 1/v on the uv-plane. If y= 1, then u= x and v= 1. Graph v= 1 on the uv-plane. If x= 0, u= 0, v= y. Graph u= 0 on the uv-plane. If y= 0, u is infinite so that does not give a bound. What region is inside those boundaries?

    Look at the boundary lines x= 1 and y= 1. On x= 1, you have u= 1/(y+1) and v= y+1. That is, u= 1/v. On y= 1, you have u= x/(x+1) and v= x+1. x= v-1 so u= (v-1)/v= 1- 1/v. Graph those curves on the u-v plane.

    Yes,
     
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