- #1
Juan Carlos
- 22
- 0
Homework Statement
I have to find the Laplace operator asociated to the next quasi-spherical curvilinear coordinates, for z>0.
Homework Equations
\begin{align}
x&=\rho \cos\phi\nonumber\\
y&=\rho \sin \phi\nonumber\\
z&=\sqrt{r^2-\rho^2},
\end{align}
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?