# Laplace-Beltrami Operator non-curvilinear coordinates

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1. Feb 15, 2016

### Juan Carlos

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

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},

and (with help of the inverse matrix and determinant) substituting in

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

\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
,
is there an easy way to check this expression?
do these coordinates have a name?

2. Feb 20, 2016