Pre-Quantum Electrodynamics

Cylindrical Coordinates c.m.cs.cyl

\((r, \phi, z)\). Relation to Cartesian coordinates:

\[ x = r \cos \phi, y = r \sin \phi, z = z \label{Gr(1.74)} \]

The unit vectors are

\[ \hat{\bf r} = \cos \phi ~\hat{\bf x} + \sin \phi~\hat{\bf y}, \hat{\boldsymbol \phi} = -\sin \theta ~\hat{\bf x} + \cos \phi~\hat{\bf y}, \hat{\bf z} = \hat{\bf z}. \label{Gr(1.75)} \]

Infinitesimal displacement:

\[ d{\bf l} = dr ~\hat{\bf r} + r d\phi~\hat{\boldsymbol \phi} + dz ~\hat{\bf z}. \label{Gr(1.77)} \]

Volume element:

\[ d\tau = r dr d\phi dz \label{Gr(1.78)} \]

Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in ]-\infty, \infty[\).

Gradient

  • Gr4(1.79)

\[ {\boldsymbol ∇} T = \frac{\partial T}{\partial r}~\hat{\bf r}

  • \frac{1}{r} \frac{\partial T}{\partial \phi}~\hat{\boldsymbol \phi}
  • \frac{\partial T}{\partial z} ~\hat{\bf z}

\tag{cylgrad}\label{cylgrad} \]

Divergence

  • Gr4(2.21)

\[ {\boldsymbol ∇} ⋅ {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r)

  • \frac{1}{r} \frac{∂ v_{φ}}{∂ φ} + \frac{\partial v_z}{\partial z}.

\tag{cyl_div} \label{cyl_div} \]

Curl

  • Gr4(2.21)

\[ {\boldsymbol ∇} × {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{∂ v_{φ}}{∂ z}\right) ~\hat{\bf r}

  • \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi}
  • \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{φ}) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z}

\tag{cyl_curl} \label{cyl_curl} \]

Laplacian

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

  • \frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2}

\label{Gr(1.82)} \]

Author: Jean-Sébastien Caux

Created: 2022-02-08 Tue 06:55

Validate