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Laplacian of polar coordinates

  1. Feb 27, 2017 #1
    1. The problem statement, all variables and given/known data
    I am trying to calculate the laplacian in polar coordinates but I failed.Please see the attached

    2. Relevant equations


    3. The attempt at a solution
    My solution to this was uploaded in the attached.I was wondering what's wrong with the purple brackets since they shouldn't exist( If you sum A,B,C and D up). Thanks
     

    Attached Files:

  2. jcsd
  3. Feb 27, 2017 #2

    kuruman

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    I don't think you should be messing with unit vectors and their derivatives. The Laplacian is a scalar operator. It looks like you are doing it in two dimensions. Start with $$\nabla ^2=\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$$
    Observe that according to the chain rule
    $$\frac{\partial}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial \theta}{\partial x}\frac{\partial}{\partial \theta}$$
    and one more time for the second partial derivative. Repeat with y and then add the two results.

    This is the "brute force" method and involves tedious algebra. It is the kind of thing that one has to do once in one's life to satisfy oneself that one can do it. :smile:
     
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