Pre-Quantum Electrodynamics

Spherical Coordinates c.m.cs.sph

In this system, we use coordinates \((r, \theta, \varphi)\) in which \(r\) is the distance from the chosen origin, \(\theta\) is the polar angle and \(\varphi\) is the azimuthal angle.

The usual Cartesian coordinates relate to spherical coordinates according to

\[ x = r \sin \theta \cos \varphi, \hspace{5mm} y = r \sin \theta \sin \varphi, \hspace{5mm} z = r \cos \theta. \tag{sph_xyz}\label{sph_xyz} \]

The unit vectors are written \(\hat{\boldsymbol r}\), \(\hat{\boldsymbol \theta}\) and \(\hat{\boldsymbol \varphi}\). A generic vector can be expressed as \[ {\bf v} = v_r \hat{\bf r} + v_{\theta} \hat{\bf \theta} + v_{\varphi} \hat{\boldsymbol \varphi} \] where the explicit relation between spherical and Cartesian unit vectors is

\begin{align} \hat{\boldsymbol r} &= \sin \theta \cos \varphi ~\hat{\bf x} + \sin \theta \sin \varphi ~\hat{\bf y} + \cos \theta ~\hat{\bf z}, \nonumber \\ \hat{\boldsymbol \theta} &= \cos \theta \cos \varphi ~\hat{\bf x} + \cos \theta \sin \varphi ~\hat{\bf y} - \sin \theta ~\hat{\bf z}, \nonumber \\ \hat{\boldsymbol \varphi} &= -\sin \varphi ~\hat{\bf x} + \cos \varphi ~\hat{\bf y}. \tag{sph_uv}\label{sph_uv} \end{align}

Do be careful: these unit vectors are direction dependent, i.e. we should really write \(\hat{\boldsymbol r} (\theta, \varphi)\), \(\hat{\boldsymbol \theta} (\theta, \varphi)\) and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).

An infinitesimal displacement \(d{\bf l}\) can be written as

\[ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin \theta d\varphi ~\hat{\boldsymbol \varphi}. \tag{sph_dl}\label{sph_dl} \]

Infinitesimal volume element:

\[ d\tau = dl_r dl_{\theta} dl_{\varphi} = r^2 \sin \theta dr d\theta d\varphi \tag{sph_dtau}\label{sph_dtau} \]

Infinitesimal surface element: depends on situation.

Gradient
\begin{equation} {\boldsymbol \nabla} T = \frac{\partial T}{\partial r} \hat{\boldsymbol r} + \frac{1}{r} \frac{\partial T}{\partial \theta} \hat{\boldsymbol \theta} + \frac{1}{r\sin \theta} \frac{\partial T}{\partial \varphi} \hat{\boldsymbol \varphi}. \tag{sph_grad}\label{sph_grad} \end{equation}
Divergence
\begin{equation} {\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 v_r) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (\sin\theta v_{\theta}) + \frac{1}{r \sin \theta} \frac{\partial v_{\varphi}}{\partial \varphi} \tag{sph_div}\label{sph_div} \end{equation}
Curl
\begin{align} {\boldsymbol \nabla} \times {\bf v} = &\frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\varphi}) - \frac{\partial v_{\theta}}{\partial \varphi} \right] \hat{\bf r} \nonumber \\ &+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \varphi} - \frac{\partial}{\partial r} (r v_{\varphi}) \right] \hat{\boldsymbol \theta} + \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \varphi} \tag{sph_curl}\label{sph_curl} \end{align}
Laplacian
\begin{equation} {\boldsymbol \nabla}^2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial T}{\partial r}\right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial T}{\partial \theta}\right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \varphi^2} \tag{sph_Lap}\label{sph_Lap} \end{equation}


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Author: Jean-Sébastien Caux

Created: 2024-02-27 Tue 10:31