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Math Help - Multivariate function Image

  1. #1
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    Multivariate function Image

    if F: reals^2 -- reals ^2 is defined by F(u,v):=(u/(1+v),uv/(1+v) =: (x(u,v),y(u,v))
    And W is the rectangle {(u,v) s.t. 0< or = u < or = 2, 0 < or equal v < or equal 1} in u-v space
    could someone describe the image F(W) in x-y space by first showing F^(-1)[(x,y)]=(x+y,y/x)=: (u(x,y),v(x,y))
    Last edited by CaptainBlack; February 10th 2011 at 09:41 PM. Reason: thread title
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  2. #2
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    Quote Originally Posted by maximus101 View Post
    if F: reals^2 -- reals ^2 is defined by F(u,v):=(u/(1+v),uv/(1+v) =: (x(u,v),y(u,v))
    And W is the rectangle {(u,v) s.t. 0< or = u < or = 2, 0 < or equal v < or equal 1} in u-v space
    could someone describe the image F(W) in x-y space by first showing F^(-1)[(x,y)]=(x+y,y/x)=: (u(x,y),v(x,y))
    First Hello and welcome to the forum. You have been posting alot of questions. I would recommend you learn LaTex. See here http://www.mathhelpforum.com/math-help/f47/

    Note you can also click on the math symbols to see the code that generated them.

    Now onto your question.

    Note that you have the system of equations.

    \displaystyle x=\frac{u}{1+v} and \displaystyle y=\frac{uv}{1+v}

    This implies that \displaystyle xv=\frac{uv}{1+v}=y \implies v=\frac{y}{x}

    Now using this we get

    \displaystyle x=\frac{u}{1+v} \iff u=(1+v)x=\left( 1+\frac{y}{x}\right)x=x+y

    Can you finish from here?
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