Poisson equation

For a general coordinate system, the metric is given by

\begin{equation} ds^2 = \sum h_{ij} dx_i dx_j\,. \end{equation}

For a coordinate system with orthogonal axes, \(h_{ij}=\delta^{\rm K}_{ij}h_{\rm i}^2\). In this case the differential vector is

\begin{equation} d\vec{x} = \sum \frac{\partial \vec{x}}{\partial q_i} dq_i \end{equation}

The unit directional vectors will be

\begin{equation} \hat{e}_i = \frac{1}{h_i} \frac{\partial \vec{x}}{\partial q_i}\,. \end{equation}

The volume elements transform with the Jacobian,

\begin{equation} d^3\vec{x} = \prod h_i dq_i\,. \end{equation}

The gradient is given by

\begin{equation} \vec\nabla \Phi = \sum \frac{1}{h_i} \frac{\partial \Phi}{\partial q_i}\hat{e}_i \end{equation}

The divergence is given by

\begin{equation} \vec\nabla. \vec{A} = \frac{1}{h_1h_2h_3} \left[ \sum_{\rm cyclic} \frac{\partial (h_2h_3 A_1)}{\partial q_1} \right] \end{equation}

The curl is given by

\begin{equation} \left.\vec\nabla \times \vec{A}\right|_3 = \frac{1}{h_1h_2} \left[ \frac{\partial (h_2A_2)}{\partial q_1} - \frac{\partial (h_1A_1)}{\partial q_2} \right] \end{equation}

The Laplacian is given by

\begin{equation} \nabla^2\Phi = \frac{1}{h_1h_2h_3} \left[ \sum_{\rm cyclic} \frac{\partial }{\partial q_1} \left(\frac{h_2h_3}{h_1}\frac{\partial\psi}{\partial q_1}\right) \right] \end{equation}

Spherical coordinate system

The metric for the spherical coordinate system is given by \begin{equation} ds^2 = dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2 \end{equation}

Thus, we have

\begin{equation} h_1 = 1 ; \,\,\, h_2 = r ;\,\,\, h_3 = r \sin \theta \,. \end{equation}

The Laplacian is then given by

\begin{equation} \nabla^2\Phi = \frac{1}{r^2}\frac{\partial }{\partial r} \left(r^2\frac{\partial \Phi}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta} \left[ \sin \theta \frac{\partial \Phi}{\partial \theta}\right] + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 \Phi}{\partial \phi^2} \end{equation}

Cylindrical coordinate system

For the cylindrical coordinate system, the metric is given by

\begin{equation} ds^2 = dR^2 + R^2 d\phi^2 + dz^2 \end{equation}

such that

\begin{equation} h_1 = 1 ; \,\,\, h_2 = R ;\,\,\, h_3 = 1 \,. \end{equation}

The Laplacian is given by

\begin{equation} \nabla^2\Phi = \frac{1}{R}\frac{\partial }{\partial R} \left(R\frac{\partial \Phi}{\partial R}\right) + \frac{1}{R^2}\frac{\partial^2}{\partial \phi^2} \Phi + \frac{\partial^2 \Phi}{\partial z^2} \end{equation}