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Question about Coordinate Change

  1. Jun 16, 2015 #1

    stevendaryl

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    Suppose that I have a two-dimensional coordinate system [itex](x,y)[/itex] and I change to a new coordinate system [itex](u,v)[/itex]. What I know is that there is some function [itex]\theta(u,v)[/itex] such that:
    1. [itex]\dfrac{\partial x}{\partial u} = cos(\theta)[/itex]
    2. [itex]\dfrac{\partial x}{\partial v} = -sin(\theta)[/itex]
    3. [itex]\dfrac{\partial y}{\partial u} = sin(\theta)[/itex]
    4. [itex]\dfrac{\partial y}{\partial v} = cos(\theta)[/itex]
    My question is: do these 4 equations imply that [itex]\theta =[/itex] constant? (so that the relationship between the coordinate systems is linear)
     
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  3. Jun 16, 2015 #2

    wabbit

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    Looks like it yes, if you compute the cross derivatives
    ## \frac{\partial^2x}{\partial u\partial v} =-\sin\theta\frac{\partial\theta}{\partial v}=-\cos\theta\frac{\partial\theta}{\partial u}\\\frac{\partial^2y}{\partial u\partial v} =\cos\theta\frac{\partial\theta}{\partial v}=-\sin\theta\frac{\partial\theta}{\partial u} ##
    you get a set of equations that require ## \frac{\partial\theta}{ \partial u}=\frac{\partial\theta}{ \partial v}=0##
     
  4. Jun 16, 2015 #3

    stevendaryl

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    Thank you! I tried exactly that, but I made a stupid sign error, and found them consistent.
     
  5. Jun 16, 2015 #4

    wabbit

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    Ah yes, signs are the spawn of the devil :)
     
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