Overview
Cylindrical Coordinates {% \rho, \phi, z %} are defined in terms of the corresponding Cartesian coordinates {% (x,y,z) %} by the following equations:
{% x = \rho cos \phi %}
{% y = \rho sin \phi %}
{% z = z %}
Vector Operators
{% \nabla = \hat{\rho} \frac{\partial}{\partial \rho} + \frac{\hat{\phi}}{\rho} \frac{\partial}{\partial \phi} + \hat{z} \frac{\partial}{\partial z} %}
{% \nabla \cdot \vec{V} = \frac{1}{\rho} \frac{\partial (\rho V_{\rho})}{\partial \rho} + \frac{1}{\rho} \frac{\partial V_\phi}{\partial \phi} + \frac{\partial V_z}{\partial z} %}
{% \nabla \times \vec{V} =
[\frac{1}{\rho} \frac{\partial V_z}{\partial \phi} - \frac{\partial V_\phi}{\partial z}] \hat{\rho}
+ [\frac{\partial V_\rho}{\partial z} - \frac{\partial V_z}{\partial \rho}] \hat{\phi}
+ \frac{1}{\rho}[\frac{\partial(\rho V_\phi)}{\partial \rho} - \frac{\partial V_\rho}{\partial \phi}] \hat{z}
%}
{% \nabla^2 A = \frac{1}{\rho} \frac{\partial}{\partial \rho}(\rho \frac{\partial A}{\partial \rho})
+ \frac{1}{\rho^2} \frac{\partial^2 A}{\partial \phi^2}
+ \frac{\partial^2 A}{\partial z^2}
%}