Pre-Quantum Electrodynamics

Spherical Coordinates c.m.cs.sph
sphere
Figure 1: Sphere drawn with Asymptote

\((r, \theta, \phi)\). \(\theta\) is the polar angle, \(\phi\) the azimuthal angle.

\[ x = r \sin \theta \cos \phi, y = r \sin \theta \sin \phi, z = r \cos \theta. \label{Gr(1.62)} \]

Unit vectors: \(\hat{\boldsymbol r}, \hat{\boldsymbol \theta}, \hat{\boldsymbol \phi}\).

\[ {\bf A} = A_r \hat{\bf r} + A_{\theta} \hat{\bf \theta} + A_{\phi} \hat{\boldsymbol \phi} \label{Gr(1.63)} \]

In terms of Cartesian unit vectors:

\begin{align} \hat{\boldsymbol r} &= \sin \theta \cos \phi \hat{\bf x} + \sin \theta \sin \phi \hat{\bf y} + \cos \theta \hat{\bf z}, \nonumber \\ \hat{\boldsymbol \theta} &= \cos \theta \cos \phi \hat{\bf x} + \cos \theta \sin \phi \hat{\bf y} - \sin \theta \hat{\bf z}, \nonumber \\ \hat{\boldsymbol \phi} &= -\sin \phi \hat{\bf x} + \cos \phi \hat{\bf y}. \label{Gr(1.64)} \end{align}

Careful: these unit vectors are direction dependent, i.e. we should really write \(\hat{\boldsymbol r} (\theta, \phi), \hat{\boldsymbol \theta} (\theta, \phi), \hat{\boldsymbol \phi} (\theta, \phi)\).

Infinitesimal displacement \(d{\bf l}\):

\[ d{\bf l} = dr \hat{\boldsymbol r} + r d\theta \hat{\boldsymbol \theta} + r\sin \theta d\phi \hat{\boldsymbol \phi}. \label{Gr(1.68)} \]

Infinitesimal volume element:

\[ d\tau = dl_r dl_{\theta} dl_{\phi} = r^2 \sin \theta dr d\theta d\phi \label{Gr(1.69)} \]

Infinitesimal surface element: depends on situation.

Gradient

\[ {\boldsymbol ∇} 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 \phi} \hat{\boldsymbol \phi}.

\label{Gr(1.70)} \]

Divergence

\[ {\boldsymbol ∇} ⋅ {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r2 vr) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (sinθ vθ)

  • \frac{1}{r \sin \theta} \frac{∂ vφ}{∂ φ}

\label{Gr(1.71)} \]

Curl

\[ {\boldsymbol ∇} × {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (sin θ vφ) - \frac{∂ vθ}{∂ φ} \right] \hat{\bf r}

  • \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi} - \frac{\partial}{\partial r} (r vφ) \right] \hat{\boldsymbol \theta}
  • \frac{1}{r} \left[ \frac{\partial}{\partial r} (r vθ) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \phi}

\label{Gr(1.72)} \]

Laplacian

\[ {\boldsymbol ∇}2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r2 \frac{\partial T}{\partial r}\right)

  • \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( sin θ \frac{\partial T}{\partial \theta}\right)
  • \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \phi^2}

\label{Gr(1.73)} \]

Author: Jean-Sébastien Caux

Created: 2022-02-07 Mon 08:02

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