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Homework Help: Multivariable partial derivative

  1. Dec 7, 2015 #1
    1. The problem statement, all variables and given/known data
    From the transformation from polar to Cartesian coordinates, show that

    \frac{\partial}{\partial x} = \cosφ \frac{\partial}{\partial r} - \frac{\sinφ}{r} \frac{\partial}{\partialφ}

    2. Relevant equations
    The transformation from polar to Cartesian coordinates is assumed to be x = r\cosφ

    3. The attempt at a solution
    To solve the problem i tried to use the multivariable chain rule. Resulting in the following equation:

    \frac{\partial}{\partial x} =\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partialφ}{\partial x}\frac{\partial}{\partial φ}

    Writing ##r = x/\cosφ## and ##\arccos(x/r) = φ## i tried to solve this problem. But this does not give the right answer.

    Am i using the right approach? I think it is necessary to use the multivariable chain rule in some form. But the partial derivative not acting on some other function seems a bit weird to me so i am not sure how to solve this problem.
    Last edited: Dec 7, 2015
  2. jcsd
  3. Dec 7, 2015 #2

    Ray Vickson

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    In LaTeX, standard functions look a lot better if they are preceded by '\', so you get ##\sin \phi## instead of ##sin \phi##, etc.
  4. Dec 7, 2015 #3


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    With that equation in mind:
    ##φ=\arctan(\frac{y}{x})## (with some subtleties).
  5. Dec 8, 2015 #4
    Ahh, thanks a lot. That solved the problem.
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