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Math Help - Change of variable for Laplace equation

  1. #1
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    Change of variable for Laplace equation

    How do I show that the change of variable:
    x = rcosθ , y = rsinθ

    converts the Laplace equation uxx + uyy = 0 to

    urr + 1/r ur + 1/r2 uθθ = 0

    Here is my understanding:

    uxx is the second partial derivative of u with respect to x, also written as d2u / d2x, but what is the equation with u in it that I have to differentiate?


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  2. #2
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    Quote Originally Posted by jkhayer View Post
    How do I show that the change of variable:





    x = rcosθ , y = rsinθ

    converts the Laplace equation uxx + uyy = 0 to
    urr + 1/r ur + 1/r2 uθθ = 0

    Here is my understanding:

    uxx is the second partial derivative of u with respect to x, also written as d2u / d2x, but what is the equation with u in it that I have to differentiate?

    Use the transformation rules

     <br />
u_x = \cos \theta u_r - \frac{\sin \theta}{r} u_{\theta},\;\;\;u_y = \sin \theta u_r + \frac{\cos \theta}{r} u_{\theta}<br />

    and the operators

     <br />
\frac{\partial}{\partial x} = \cos \theta \frac{\partial}{\partial r} - \frac{\sin \theta}{r} \frac{\partial}{\partial \theta},\;\;\;\frac{\partial}{\partial y} = \sin \theta \frac{\partial}{\partial r} + \frac{\cos \theta}{r} \frac{\partial}{\partial \theta}.<br />

    Find \frac{\partial}{\partial x} \left(u_x \right) and similarly for \frac{\partial}{\partial y} \left(u_y \right) using the above, add and simplify.
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