math lessons - Nabla in cylindrical and spherical coordinates

This is a list of some vector calculus formulae of general use in working with standard coordinate systems.

**Table with the del or nabla in cylindrical and spherical coordinates**
Operation	Cartesian coordinates (x,y,z)	Cylindrical coordinates (ρ,φ,z)	Spherical coordinates (r,θ,φ)
Definition of coordinates		$\left[\begin{matrix} x & = & \rho\cos\phi \\ y & = & \rho\sin\phi \\ z & = & z \end{matrix}\right.$	$\left[\begin{matrix} x & = & r\sin\theta\cos\phi \\ y & = & r\sin\theta\sin\phi \\ z & = & r\cos\theta \end{matrix}\right.$
Definition of coordinates		$\left[\begin{matrix} \rho & = & \sqrt{x^2 + y^2} \\ \phi & = & \operatorname{atan2}(y, x) \\ z & = & z \end{matrix}\right.$	$\left[\begin{matrix} r & = & \sqrt{x^2 + y^2 + z^2} \\ \theta & = & \arccos(z / r) \\ \phi & = & \operatorname{atan2}(y, x) \end{matrix}\right.$
$\mathbf{A}$	$A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}$	$A_\rho\boldsymbol{\hat \rho} + A_\phi\boldsymbol{\hat \phi} + A_z\boldsymbol{\hat z}$	$A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}$
$\nabla f$	${\partial f \over \partial x}\mathbf{\hat x} + {\partial f \over \partial y}\mathbf{\hat y} + {\partial f \over \partial z}\mathbf{\hat z}$	${\partial f \over \partial \rho}\boldsymbol{\hat \rho} + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat \phi} + {\partial f \over \partial z}\boldsymbol{\hat z}$	${\partial f \over \partial r}\boldsymbol{\hat r} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat \theta} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat \phi}$
$\nabla \cdot \mathbf{A}$	${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$	${1 \over \rho}{\partial \rho A_\rho \over \partial \rho} + {1 \over \rho}{\partial A_\phi \over \partial \phi} + {\partial A_z \over \partial z}$	${1 \over r^2}{\partial r^2 A_r \over \partial r} + {1 \over r\sin\theta}{\partial A_\theta\sin\theta \over \partial \theta} + {1 \over r\sin\theta}{\partial A_\phi \over \partial \phi}$
$\nabla \times \mathbf{A}$	$\begin{matrix} ({\partial A_z \over \partial y} - {\partial A_y \over \partial z}) \mathbf{\hat x} & + \\ ({\partial A_x \over \partial z} - {\partial A_z \over \partial x}) \mathbf{\hat y} & + \\ ({\partial A_y \over \partial x} - {\partial A_x \over \partial y}) \mathbf{\hat z} & \ \end{matrix}$	$\begin{matrix} ({1 \over \rho}{\partial A_z \over \partial \phi} - {\partial A_\phi \over \partial z}) \boldsymbol{\hat \rho} & + \\ ({\partial A_\rho \over \partial z} - {\partial A_z \over \partial \rho}) \boldsymbol{\hat \phi} & + \\ {1 \over \rho}({\partial \rho A_\phi \over \partial \rho} - {\partial A_\rho \over \partial \phi}) \boldsymbol{\hat z} & \ \end{matrix}$	$\begin{matrix} {1 \over r\sin\theta}({\partial A_\phi\sin\theta \over \partial \theta} - {\partial A_\theta \over \partial \phi}) \boldsymbol{\hat r} & + \\ ({1 \over r\sin\theta}{\partial A_r \over \partial \phi} - {1 \over r}{\partial r A_\phi \over \partial r}) \boldsymbol{\hat \theta} & + \\ {1 \over r}({\partial r A_\theta \over \partial r} - {\partial A_r \over \partial \theta}) \boldsymbol{\hat \phi} & \ \end{matrix}$
$\Delta f = \nabla^2 f$	${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$	${1 \over \rho}{\partial \over \partial \rho}(\rho {\partial f \over \partial \rho}) + {1 \over \rho^2}{\partial^2 f \over \partial \phi^2} + {\partial^2 f \over \partial z^2}$	${1 \over r^2}{\partial \over \partial r}(r^2 {\partial f \over \partial r}) + {1 \over r^2\sin\theta}{\partial \over \partial \theta}(\sin\theta {\partial f \over \partial \theta}) + {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \phi^2}$
$\Delta \mathbf{A} = \nabla^2 \mathbf{A}$	$\mathbf{\hat x}\Delta A_x + \mathbf{\hat y}\Delta A_y + \mathbf{\hat z}\Delta A_z$	$\begin{matrix} \boldsymbol{\hat\rho}(\Delta A_\rho - {A_\rho \over \rho^2} - {2 \over \rho^2}{\partial A_\phi \over \partial \phi}) & + \\ \boldsymbol{\hat\phi}(\Delta A_\phi - {A_\phi \over \rho^2} + {2 \over \rho^2}{\partial A_\rho \over \partial \phi}) & + \\ \boldsymbol{\hat z} \Delta A_z & \ \end{matrix}$	$\begin{matrix} \boldsymbol{\hat r} & (\Delta A_r - {2 A_r \over r^2} - {2 A_\theta\cos\theta \over r^2\sin\theta} \\ \ & - {2 \over r^2}{\partial A_\theta \over \partial \theta} - {2 \over r^2\sin\theta}{\partial A_\phi \over \partial \phi}) & + \\ \boldsymbol{\hat\theta} & (\Delta A_\theta - {A_\theta \over r^2\sin^2\theta} \\ \ & + {2 \over r^2}{\partial A_r \over \partial \theta} - {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\phi \over \partial \phi}) & + \\ \boldsymbol{\hat\phi} & (\Delta A_\phi - {A_\phi \over r^2\sin^2\theta} \\ \ & + {2 \over r^2\sin^2\theta}{\partial A_r \over \partial \phi} + {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\theta \over \partial \phi}) & \ \end{matrix}$
Differential Displacement	$d\mathbf{l} = dx\mathbf{\hat x} + dy\mathbf{\hat y} + dz\mathbf{\hat z}$	$d\mathbf{l} = d\rho\boldsymbol{\hat \rho} + \rho d\phi\boldsymbol{\hat \phi} + dz\boldsymbol{\hat z}$	$d\mathbf{l} = dr\mathbf{\hat r} + rd\theta\boldsymbol{\hat \theta} + r\sin\theta d\phi\boldsymbol{\hat \phi}$
Differential Normal Area	$\begin{matrix}d\mathbf{S} = &dydz\mathbf{\hat x} \\ &dxdz\mathbf{\hat y} \\ &dxdy\mathbf{\hat z}\end{matrix}$	$\begin{matrix} d\mathbf{S} = & \rho d\phi dz\boldsymbol{\hat \rho} \\ & d\rho dz\boldsymbol{\hat \phi} \\ & \rho d\rho d\phi \mathbf{z} \end{matrix}$	$\begin{matrix} d\mathbf{S} = & r^2 \sin\theta d\theta d\phi \mathbf{\hat r} \\ & r\sin\theta drd\phi \boldsymbol{\hat \theta} \\ & rdrd\theta\boldsymbol{\hat \phi} \end{matrix}$
Differential Volume	$dv = dxdydz\,\!$	$dv = \rho d\rho d\phi dz\,\!$	$dv = r^2\sin\theta drd\theta d\phi\,\!$
Non-trivial calculation rules: $\operatorname{div\ grad\ } f = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f$ (Laplacian) $\operatorname{curl\ grad\ } f = \nabla \times (\nabla f) = 0$ $\operatorname{div\ curl\ } \mathbf{A} = \nabla \cdot (\nabla \times \mathbf{A}) = 0$ $\operatorname{curl\ curl\ } \mathbf{A} = \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ $\Delta f g = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f$ Lagrange's formula for the cross product: $\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B} (\mathbf{A} \cdot \mathbf{C}) - \mathbf{C} (\mathbf{A} \cdot \mathbf{B})$