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I Derivative with several terms in denominator

  1. Feb 6, 2017 #1
    Hi. I want to solve [itex]\frac{\partial x^{\nu}}{\partial x^{\mu} + \xi ^{\mu}}[/itex], knowing that [itex]\frac{\partial x^{\nu}}{\partial x^{\mu}} = \delta ^{\nu}_{\mu}[/itex]. How can I do this?
     
  2. jcsd
  3. Feb 6, 2017 #2

    andrewkirk

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    What do ##\partial x^\nu## and ##\partial x^\mu## denote?
     
  4. Feb 6, 2017 #3
    [itex]\partial[/itex] is the symbol for partial derivative and [itex]x^{\rho}[/itex] is the coordinate of a point [itex]x[/itex].
     
  5. Feb 6, 2017 #4

    andrewkirk

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    In that case the expression in the OP has no meaning. It is simply a misuse of the partial derivative symbol.
     
  6. Feb 6, 2017 #5
    No. It is supposed to be a derivative. I must evaluate the derivative of [itex]x^{\nu}[/itex] with respect to [itex]x^{\mu}+ \xi^{\mu}[/itex].
     
  7. Feb 6, 2017 #6

    andrewkirk

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    In that case, there are necessary parentheses missing in the OP. It needs to be written
    $$\frac{\partial x^\nu}{\partial (x^\mu+\xi^\mu)}$$
    and ##x^\nu## needs to be specified as a function of ##x^\mu+\xi^\mu##. What is that function? Perhaps if you provided more information about the context of your question, the function would become apparent.
     
  8. Feb 6, 2017 #7

    Stephen Tashi

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  9. Feb 6, 2017 #8
    I didn't notice any two or more terms in the denominator of those derivatives.

    Yes. I'm sorry.

    I was trying to relate the components of a vector in the new ##x## coordinates with that in the ##y## coordinates. They should change as $$V^{\nu}(x) = \frac{\partial x^{\nu}}{\partial (y^{\mu} = x^{\mu}+ \epsilon \xi^{\mu}(x))}V'^{\mu}(y).$$

    There was missing the ##\epsilon## (|##\epsilon##| << 1) in the OP, because points ##y## and ##x## are very close from each other.

    I found the solution for this derivative in books of GR. It involves expanding something, where one gets terms in higher orders in ##\epsilon##, there was also a minus sign. But can't remember more than this...
     
    Last edited: Feb 6, 2017
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