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Math Help - Laplacian Operator

  1. #1
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    Laplacian Operator

    Here is a exercise that our teacher gave us. He said to try it out. I don't understand it. Help would be greatly appreciated.

    Thanks
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Suppose you have a bijective, analytic mapping f:\mathbb{R}^3 \rightarrow \mathbb{R}^3, given by

    (x,y,z) \mapsto (r(x,y,z), s(x,y,z), t(x,y,z)).

    Then

    \frac{\partial}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial }{\partial r}+\frac{\partial s}{\partial x}\frac{\partial }{\partial s}+\frac{\partial t}{\partial x}\frac{\partial }{\partial t}


    \frac{\partial}{\partial y} = \frac{\partial r}{\partial y}\frac{\partial }{\partial r}+\frac{\partial s}{\partial y}\frac{\partial }{\partial s}+\frac{\partial t}{\partial y}\frac{\partial }{\partial t}


    \frac{\partial}{\partial z} = \frac{\partial r}{\partial z}\frac{\partial }{\partial r}+\frac{\partial s}{\partial z}\frac{\partial }{\partial s}+\frac{\partial t}{\partial z}\frac{\partial }{\partial t}


    Apply this theorem (which is the chain rule in 3 dimensions) to the cylindrical coordinate transformation.
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