Cylindrical Coordinates

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Cylindrical coordinates are curvilinear) three dimensional coordinates. One variable gives the position along the axis of a cylinder, the others give the distance away from the axis, and the angle (relative to some base) which this point is rotated by.

There exist numerous notations. The plasmagicians convention uses variables labelled r,φ, and z.

With respect to Cartesian Coordinates

x = rcosφ,y = rsinφ,z = z
r = \sqrt{x^2 + y^2}, \phi = \tan^{-1}{\frac{y}{x}}, z = z

Or, if you dislike tan − 1,

\phi = \textbf{atan2}(y,x)

Where atan2 is substantially more sensible.

The Helmholtz Equation (and its subcase, the Laplace equation) is separable in Cylindrical Coordinates.

Commonly Used Factors and Operators
Value \mathbf{\hat{r}} \mathbf{\hat{\phi}} \mathbf{\hat{z}}
Scale Factors 1 r 1
Derivatives of the Identity Vector <\mathbf{r},\mathbf{\phi},\mathbf{z}> \frac{\partial \mathbf{\hat{r}}}{\partial \phi} = \mathbf{\hat{\phi}} \frac{\partial \mathbf{\hat{\phi}}}{\partial \phi} = -\mathbf{\hat{r}} 0
Gradient, \mathbf{\nabla} \frac{\partial}{\partial r} \frac{1}{r}\frac{\partial}{\partial \phi} \frac{\partial}{\partial z}
Laplacian, \nabla^2 f \frac{1}{r}\frac{\partial}{\partial r} \left({r \frac{\partial f}{\partial r}}\right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2}
Vector Laplacian, \mathbf{\nabla^2} \mathbf{F} Honestly too awful to transcribe.
Divergence, \mathbf{\nabla \cdot F} \frac{1}{r}\frac{\partial}{\partial r} \left({r F_r}\right) + \frac{1}{r} \frac{\partial \mathbf{F}_\phi}{\partial \phi} + \frac{\partial \mathbf{F}_z}{\partial z}
Curl, \mathbf{\nabla \times F} \left|{\begin{bmatrix} \frac{\mathbf{\hat{r}}}{r} & \mathbf{\hat{\phi}} & \frac{\mathbf{\hat{z}}}{r} \\ \\ {\frac{\partial}{\partial r}} & {\frac{\partial}{\partial \phi}} & {\frac{\partial}{\partial z}} \\ \\ F_r & r F_\phi & F_z \end{bmatrix}}\right| \frac{1}{r} \frac{\partial \mathbf{F}_z}{\partial \phi} - \frac{\partial \mathbf{F}_\phi}{\partial z} \frac{\partial \mathbf{F}_r}{\partial z} - \frac{\partial \mathbf{F}_z}{\partial r} \frac{1}{r} \left[{\frac{\partial}{\partial r} \left({r F_\phi}\right) - \frac{\partial \mathbf{F}_r}{\partial \phi}}\right]
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