Pre-Quantum Electrodynamics
Cylindrical Coordinatesc.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 \(z \in ]-\infty, \infty[\).
Curl
\[ {\boldsymbol ∇} × {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{∂ vφ}{∂ z}\right) ~\hat{\bf r}
- \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi}
- \frac{1}{r} \left( \frac{\partial}{\partial r} (r vφ) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z}
\tag{cyl_curl} \label{cyl_curl} \]
Laplacian
\[ {\boldsymbol ∇}2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right)
- \frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2}
\label{Gr(1.82)} \]
Created: 2022-02-07 Mon 08:02