$$\require{physics}$$ Vector Field $$ \vb{E} = E_{r} \vectorunit{r} + E_{\phi} \vectorunit{\phi} + E_{z} \vectorunit{z} $$ 1. Gradient $$ \grad{V} = \pdv{V}{r} \vectorunit{r} + \frac{1}{r} \pdv{V}{\phi} \vectorunit{\phi} + \pdv{V}{z} \vectorunit{z} $$ 2. Divergence $$ \divergence{ \vb{E} } = \frac{1}{r} \pdv{ (r E_{r})}{r} + \frac{1}{r} \pdv{E_{\phi}}{\phi} + \pdv{E_{z}}{z} $$ 3. Curl $$ \curl{ \vb{E} } = \frac{1}{r} \begin{vmatrix} \vectorunit{r} & r \vectorunit{\phi} & \vectorunit{z} \\ \pdv{}{r} & \pdv{}{\phi} & \pdv{}{z} \\ E_{r} & r E_{\phi} & E_{z} \\ \end{vmatrix} $$ $$ \frac{1}{r} \begin{vmatrix} \vectorunit{r} & r \vectorunit{\phi} & \vectorunit{z} \\ \pdv{}{r} & \pdv{}{\phi} & \pdv{}{z} \\ E_{r} & r E_{\phi} & E_{z} \\ \end{vmatrix} = \left( \frac{1}{r} \pdv{E_{z}}{\phi} - \pdv{E_{\phi}}{z} \right) \vectorunit{r} + \left( \pdv{E_{r}}{z} - \pdv{E_{z}}{r} \right) \vectorunit{\phi} + \frac{1}{r} \left( \pdv{ (r E_{\phi} ) }{r} - \pdv{E_{r}}{\phi} \right) \vectorunit{z} $$ 4. Laplacian $$ \laplacian{ f } = \divergence{ \grad{ f }} $$ $$ \laplacian{ f } = \frac{1}{r} \pdv{ }{r} \left( r \pdv{f}{r} \right) + \frac{1}{r^2} \pdv[2]{f}{\phi} + \pdv[2]{f}{z} $$ Sometimes the Laplacian is written this way: $$ \laplacian{ f } = \pdv[2]{f}{r} + \frac{1}{r} \pdv{f}{r} + \frac{1}{r^2} \pdv[2]{f}{\phi} + \pdv[2]{f}{z} $$ 5. Wave Equation $$ \frac{1}{c^2} \pdv[2]{f}{t} = \laplacian{f} $$ $$ \frac{1}{c^2} \pdv[2]{f}{t} = \pdv[2]{f}{r} + \frac{1}{r} \pdv{f}{r} + \frac{1}{r^2} \pdv[2]{f}{\phi} + \pdv[2]{f}{z} $$ 6. Bessel Function. Bessel Functions of integer order are solutions to the Wave Equation in Cylindrical Coordinates. They are also, naturally, solutions to Bessel's Equation. Here are a couple different ways of writing Bessel's Equation: $$ x^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} + x \frac{d}{d x} y{\left(x \right)} + \left( x^{2} - n^{2} \right) y{\left(x \right)} = 0 $$ $$ x^{2} \frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + \left( x^{2} - n^{2} \right) y = 0 $$ $$ x^{2} y'' + x y' + \left( x^{2} - n^{2}\right) y = 0 $$ The solutions are combinations of Bessel Functions of the First and Second kind. $$ y{\left(x \right)} = C_{1} J_{n}\left(x\right) + C_{2} Y_{n}\left(x\right) $$ Bessel Function of the First Kind: $$ J_n(x)=\frac{1}{\pi}\int_0^\pi\cos(x\sin \theta-n\theta)\mathrm{d}\theta $$ Another way of defining the Bessel Function of the first kind, where n is an integer: $$ J_n(x) = \sum_{k=0}^{\infty} \frac{ (-1)^{k} }{ k! \; (n+k)! }\left( \frac{x}{2} \right)^{n+2k} $$ Examples and Python code coming soon...
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