Circular cylindrical coordinates

The circular cylindrical coordinates $\rho, \varphi, z$ are related to the rectangular Cartesian coordinates $x$,$y$, $z$ by the formulas (see Fig. A.4):

Figure A.4: Circular cylindrical coordinates.

$$\left\{ \begin{array}{l} x=\rho \cos \varphi\space , \\ y=\rho \... ...ce ,\quad 0 \le \varphi\leq 2 \pi \space , \quad -\infty < z < \infty) \right .$$ (2.17)

The inverse relations are:

$$\rho=\sqrt{x^2+y^2} \space , \quad \tan \varphi= \frac{y}{x} \space , \quad z=z \space .$$ (2.18)

An infinitesimal length $dl$ is

$$dl=\sqrt{(d\rho)^2 + (\rho \, d\varphi)^2 + (dz)^2} \space .$$ (2.19)

An infinitesimal area $dS$ is easily calculated if one of the coordinates is constant on $dS$, by taking the product of two appropriate length elements; thus:

$$dS =\rho \, d\rho \, d\varphi\space , \mbox{if $z$\space is constant,}$$ (2.20)

$$=\rho \, d\rho \, dz \space , \mbox{if $\rho$\space is constant,}$$ (2.21)

$$= d\rho \, dz \space , \mbox{if $\varphi$\space is constant.}$$ (2.22)

An infinitesimal volume element is:

$$dv=\rho \, d\rho \, d\varphi\, dz \space .$$ (2.23)

The gradient, divergence, curl and laplacian become, in cylindrical coordinates:

$$\nabla f = \hat{\rho} \frac{\partial f}{\partial \rho} + \hat{\varphi} \fr... ...{\partial f}{\partial \varphi} + \hat{z} \frac{\partial f}{\partial z} \space ,$$ (2.24)

$$\nabla \cdot \underline{F} = \frac{1}{\rho} \frac{\partial}{\partial \rho} (\rho F_{\rho}) +... ...rtial F_{\varphi}}{\partial \varphi} + \frac{\partial F_z}{\partial z} \space ,$$ (2.25)

$$\nabla \times \underline{F} = \hat{\rho} \left( \frac{1}{\rho} \frac{\partial F_z}{\partial \... ...rac{\partial F_{\rho}}{\partial z} - \frac{\partial F_z}{\partial \rho} \right)$$

$$+ \hat{z} \left( \frac{1}{\rho} \frac{\partial}{\partial \rho} (\... ..._{\varphi}) - \frac{1}{\rho} \frac{F_{\rho}}{\partial \varphi} \right) \space ,$$ (2.26)

$$\nabla^2 f = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left( \rho \frac... ...partial^2 f}{\partial \varphi^2 } + \frac{\partial ^2 f}{\partial z^2} \space ,$$ (2.27)

where

$$\underline{F}=\hat{\rho} F_{\rho} + \hat{\varphi} F_{\varphi} + \hat{z} F_z \space .$$ (2.28)