The ∇-operator
In cartesian coordinates (x,y,z) : →∇=∂∂x→ex+∂∂y→ey+∂∂z→ez , gradf=→∇f=∂f∂x→ex+∂f∂y→ey+∂f∂z→ez div →a=→∇⋅→a=∂ax∂x+∂ay∂y+∂az∂z , ∇2f=∂2f∂x2+∂2f∂y2+∂2f∂z2 rot →a=→∇×→a=(∂az∂y−∂ay∂z)→ex+(∂ax∂z−∂az∂x)→ey+(∂ay∂x−∂ax∂y)→ez In cylinder coordinates (r,φ,z) holds: →∇=∂∂r→er+1r∂∂φ→eφ+∂∂z→ez , gradf=∂f∂r→er+1r∂f∂φ→eφ+∂f∂z→ez div →a=∂ar∂r+arr+1r∂aφ∂φ+∂az∂z , ∇2f=∂2f∂r2+1r∂f∂r+1r2∂2f∂φ2+∂2f∂z2 rot →a=(1r∂az∂φ−∂aφ∂z)→er+(∂ar∂z−∂az∂r)→eφ+(∂aφ∂r+aφr−1r∂ar∂φ)→ez
In spherical coordinates (r,\theta,\varphi): \begin{aligned} \vec{\nabla} &=&\frac{\partial }{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial }{\partial \theta}\vec{e}_{\theta}+\frac{1}{r\sin\theta}\frac{\partial }{\partial \varphi}\vec{e}_{\varphi}\\ {\rm grad}f &=&\frac{\partial f}{\partial r}\vec{e}_{r}+\frac{1}{r}\frac{\partial f}{\partial \theta}\vec{e}_{\theta}+\frac{1}{r\sin\theta}\frac{\partial f}{\partial \varphi}\vec{e}_{\varphi}\\ {\rm div}~\vec{a}&=&\frac{\partial a_r}{\partial r}+\frac{2a_r}{r}+\frac{1}{r}\frac{\partial a_\theta}{\partial \theta}+\frac{a_\theta}{r\tan\theta}+\frac{1}{r\sin\theta}\frac{\partial a_\varphi}{\partial \varphi}\\ {\rm rot}~\vec{a}&=&\left(\frac{1}{r}\frac{\partial a_\varphi}{\partial \theta}+\frac{a_\theta}{r\tan\theta}-\frac{1}{r\sin\theta}\frac{\partial a_\theta}{\partial \varphi}\right)\vec{e}_{r}+ \left(\frac{1}{r\sin\theta}\frac{\partial a_r}{\partial \varphi}-\frac{\partial a_\varphi}{\partial r}-\frac{a_\varphi}{r}\right)\vec{e}_{\theta}+\\ &&\left(\frac{\partial a_\theta}{\partial r}+\frac{a_\theta}{r}-\frac{1}{r}\frac{\partial a_r}{\partial \theta}\right)\vec{e}_{\varphi}\\ \nabla^2 f &=&\frac{\partial^2 f}{\partial r^2}+\frac{2}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2}+\frac{1}{r^2\tan\theta}\frac{\partial f}{\partial \theta}+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \varphi^2}\end{aligned}
General orthonormal curvelinear coordinates (u,v,w) can be obtained from cartesian coordinates by the transformation \vec{x}=\vec{x}(u,v,w). The unit vectors are then given by: \vec{e}_{u}=\frac{1}{h_1}\frac{\partial \vec{x}}{\partial u}~,~~\vec{e}_{v}=\frac{1}{h_2}\frac{\partial \vec{x}}{\partial v}~,~~ \vec{e}_{w}=\frac{1}{h_3}\frac{\partial \vec{x}}{\partial w} where the factors h_i set the norm to 1. Then holds: \begin{aligned} {\rm grad}f &=&\frac{1}{h_1}\frac{\partial f}{\partial u}\vec{e}_{u}+\frac{1}{h_2}\frac{\partial f}{\partial v}\vec{e}_{v}+\frac{1}{h_3}\frac{\partial f}{\partial w}\vec{e}_{w}\\ {\rm div}~\vec{a}&=&\frac{1}{h_1h_2h_3}\left(\frac{\partial }{\partial u}(h_2h_3a_u)+\frac{\partial }{\partial v}(h_3h_1a_v)+\frac{\partial }{\partial w}(h_1h_2a_w)\right)\\ {\rm rot}~\vec{a}&=&\frac{1}{h_2h_3}\left(\frac{\partial (h_3a_w)}{\partial v}-\frac{\partial (h_2a_v)}{\partial w}\right)\vec{e}_{u}+ \frac{1}{h_3h_1}\left(\frac{\partial (h_1a_u)}{\partial w}-\frac{\partial (h_3a_w)}{\partial u}\right)\vec{e}_{v}+\\ &&\frac{1}{h_1h_2}\left(\frac{\partial (h_2a_v)}{\partial u}-\frac{\partial (h_1a_u)}{\partial v}\right)\vec{e}_{w}\\ \nabla^2 f &=&\frac{1}{h_1h_2h_3}\left[\frac{\partial }{\partial u}\left(\frac{h_2h_3}{h_1}\frac{\partial f}{\partial u}\right)+ \frac{\partial }{\partial v}\left(\frac{h_3h_1}{h_2}\frac{\partial f}{\partial v}\right)+ \frac{\partial }{\partial w}\left(\frac{h_1h_2}{h_3}\frac{\partial f}{\partial w}\right)\right]\end{aligned}
