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Mapping with change of variable

  1. Jul 5, 2009 #1
    For an infinitesimal mapping with u = 1,2,3,4:

    [tex]x ^u \rightarrow x^u + \xi^u(x) [/tex]

    Now suppose we introduce a new set of variables:

    [tex]x^{'u} = x^{'u}(x)[/tex]

    I would have thought the infinitesimal mapping in terms of the new variables should be written as:

    [tex] \xi^{'u}(x^{'}) = \frac{\partial \xi^{u}(x)}{\partial x^p} x^{'p} (x)[/tex]

    However, it is written as:

    [tex] \xi^{'u}(x^{'}) = \frac{\partial x^{'u}}{\partial x^u} \xi^{u} (x)[/tex]

    Does this look correct to you?
     
  2. jcsd
  3. Jul 6, 2009 #2

    tiny-tim

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    Hi jason12345! :smile:

    The infinitesimal transformation is linear, and is essentially a matrix:

    y = (I + Z)x
    y' = (I + Z')x'

    The coordinates of Z transform as Z' = (Jacobian)Z :wink:
     
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