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Laplace-Beltrami Operator non-curvilinear coordinates

  1. Feb 15, 2016 #1
    1. The problem statement, all variables and given/known data
    I have to find the Laplace operator asociated to the next quasi-spherical curvilinear coordinates, for z>0.

    2. Relevant equations

    \begin{align}

    x&=\rho \cos\phi\nonumber\\

    y&=\rho \sin \phi\nonumber\\

    z&=\sqrt{r^2-\rho^2},

    \end{align}

    3. The attempt at a solution

    I computed the metric tensor


    \begin{equation}

    g_{ij}=\begin{bmatrix}

    \dfrac{r^2}{r^2-\rho^2} & \dfrac{r\rho}{\rho^2-r^2}& 0\\

    \dfrac{r\rho}{\rho^2-r^2} & \dfrac{r^2}{r^2-\rho^2} &0 \\

    0 & 0 & \rho^2 \\

    \end{bmatrix},
    \end{equation}
    and (with help of the inverse matrix and determinant) substituting in

    \begin{equation}
    \nabla^2=\dfrac{1}{\sqrt{|g|}}\partial_i\left(\sqrt{|g|}g^{ij}\partial_j \,\right),
    \end{equation},
    explicitly

    \begin{equation}

    \nabla^2=\partial_\rho^2+\dfrac{1}{\rho}\partial_\rho+\dfrac{2\rho}{r}\partial_{\rho r}+\partial_{r}^2+\dfrac{2}{r}\partial_r+\dfrac{1}{\rho^2}\partial_{\phi}^2
    \end{equation},
    is there an easy way to check this expression?
    do these coordinates have a name?
     
  2. jcsd
  3. Feb 20, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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