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Math Help - Jacobian transformation from regions

  1. #1
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    Jacobian transformation from regions

    for the Jacobian transformation from a region S to R does this limit expression represent the slope of the tangent line to the side in the transformed region R?
    so you can approximate the area of the new region R by the cross product parallelogram since in the limit that delta u goes to 0, the curves of a parellelogram becomes straight lines? but it seems that this would only work in specific cases where region S or R is similar to a parallelogram?

    [latex]r_{u}^{} =lim_{\Delta u\rightarrow 0}\frac{r(u_o +\Delta u,{v_{o}})-r(u_{o},v_{o})}{\Delta u}[/latex]

    according to my vertice method, you don't have to transform the lines of a trapezoid into a new trapezoid, you can just calculate the vertices

    I should look at Spivack or Hubbard- I found a copy of the Loomis text. thanks very much!
    (btw how do you get the Latex to post directly on the forum?)
    (I'm having bad health problems)
    Attached Thumbnails Attached Thumbnails Jacobian transformation from regions-jacobian.gif  
    Last edited by mathnerd15; April 2nd 2013 at 10:12 PM.
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  2. #2
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    Re: Jacobian transformation between regions

    in the example given in Stewart, the straight side parallelogram is transformed into a new region with curved sides with tangent vectors. am I missing something in the Jacobian transformation? when do you use the other method of calculating the boundaries of the integral by imagining the new u,v function entering and exiting the xy region like for example a family of hyperbolas in a rectangle xy?
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