Editing Cylindrical coordinate system (section)

==Line and volume elements==
{{hatnote|1= See [[Multiple integral#Cylindrical coordinates|multiple integral]] for details of volume integration in cylindrical coordinates, and [[Del in cylindrical and spherical coordinates]] for [[vector calculus]] formulae.}}

In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes.

The [[line element]] is
<math display="block">\mathrm{d}\boldsymbol{r} = \mathrm{d}\rho\,\boldsymbol{\hat{\rho}} + \rho\,\mathrm{d}\varphi\,\boldsymbol{\hat{\varphi}} + \mathrm{d}z\,\boldsymbol{\hat{z}}.</math>

The [[volume element]] is
<math display="block">\mathrm{d}V = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}z.</math>

The [[Differential (infinitesimal)|surface element]] in a surface of constant radius {{mvar|ρ}} (a vertical cylinder) is
<math display="block">\mathrm{d}S_\rho = \rho\,\mathrm{d}\varphi\,\mathrm{d}z.</math>

The surface element in a surface of constant azimuth {{mvar|φ}} (a vertical half-plane) is
<math display="block">\mathrm{d}S_\varphi = \mathrm{d}\rho\,\mathrm{d}z.</math>

The surface element in a surface of constant height {{mvar|z}} (a horizontal plane) is
<math display="block">\mathrm{d}S_z = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi.</math>

The [[del]] operator in this system leads to the following expressions for [[gradient]], [[divergence]], [[curl (mathematics)|curl]] and [[Laplacian]]:
<math display="block">\begin{align}
  \nabla f &= \frac{\partial f}{\partial \rho}\boldsymbol{\hat{\rho}} + \frac{1}{\rho}\frac{\partial f}{\partial \varphi}\boldsymbol{\hat{\varphi}} + \frac{\partial f}{\partial z}\boldsymbol{\hat{z}} \\[8px]

  \nabla \cdot \boldsymbol{A} &= \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho A_\rho\right) + \frac{1}{\rho} \frac{\partial A_\varphi}{\partial \varphi} + \frac{\partial A_z}{\partial z} \\[8px]

  \nabla \times \boldsymbol{A} &= \left(\frac{1}{\rho}\frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z}\right)\boldsymbol{\hat{\rho}} +
    \left(\frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho}\right)\boldsymbol{\hat{\varphi}} +
    \frac{1}{\rho}\left(\frac{\partial}{\partial \rho}\left(\rho A_\varphi\right) - \frac{\partial A_\rho}{\partial \varphi}\right) \boldsymbol{\hat{z}} \\[8px]

  \nabla^2 f &= \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial f}{\partial \rho}\right) +
    \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2}
\end{align}</math>