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Math Help - Laplacian Differentiation

  1. #1
    Super Member Deadstar's Avatar
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    Laplacian Differentiation

    Let u(x, y) be a twice differentiable function. Show that in polar coordinates  x = r \cos(\theta), y = r \sin(\theta) the Laplacian of u takes the form...
    u_{xx} + u_{yy} = u_{rr} + \frac{1}{r} u_{r} + \frac{1}{r^2} \frac{{\partial}^2 u}{{\partial} {\theta}^2}.

    Should be an easy Q but i cant remember how to do the second derivative.

    u_{x} = u_{r} \frac{\partial r}{\partial x} + u_{\theta} \frac{\partial \theta}{\partial x} = u_{r} \frac{x}{r} + u_{\theta} \frac{-y}{r^2} .

    But then how do you take the second derivative?
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  2. #2
    MHF Contributor Calculus26's Avatar
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    =


    replace x = rcos(theta) y = rsin(theta)


    This won't be pleasant but:

    Uxx = (Ux)x so in replace U with Ux

    similarly for Uyy
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