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