Del in cylindrical and spherical coordinates

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This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Notes

This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and ϕ):
- The polar angle is denoted by θ: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
- The azimuthal angle is denoted by ϕ: it is the angle between the x-axis and the projection of the radial vector onto the xy-plane.
The function atan2(y, x) can be used instead of the mathematical function arctan(y/x) owing to its domain and image. The classical arctan function has an image of (−π/2, +π/2), whereas atan2 is defined to have an image of (−π, π].

Formulae

Table with the del operator in cylindrical, spherical and parabolic cylindrical coordinates
Operation	Cartesian coordinates $(x, y, z)$	Cylindrical coordinates $(ρ, ϕ, z)$	Spherical coordinates $(r, θ, ϕ)$	Parabolic cylindrical coordinates $(σ, τ, z)$
Definition of coordinates	$\begin{align} \rho &= \sqrt{x^2+y^2} \\ \phi &= \arctan(y/x) \\ z &= z \end{align}$	$\begin{align} x &= \rho\cos\phi \\ y &= \rho\sin\phi \\ z &= z \end{align}$	$\begin{align} x &= r\sin\theta\cos\phi \\ y &= r\sin\theta\sin\phi \\ z &= r\cos\theta \end{align}$	$\begin{align} x &= \sigma \tau\\ y &= \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\ z &= z \end{align}$
Definition of coordinates	$\begin{align} r &= \sqrt{x^2+y^2+z^2} \\ \theta &= \arccos(z/r)\\ \phi &= \arctan(y/x) \end{align}$	$\begin{align} r &= \sqrt{\rho^2 + z^2} \\ \theta &= \arctan{(\rho/z)}\\ \phi &= \phi \end{align}$	$\begin{align} \rho &= r\sin\theta \\ \phi &= \phi\\ z &= r\cos\theta \end{align}$	$\begin{align} \rho\cos\phi &= \sigma \tau\\ \rho\sin\phi &= \tfrac{1}{2} \left( \tau^{2} - \sigma^{2} \right) \\ z &= z \end{align}$
Definition of unit vectors	$\begin{align} \boldsymbol{\hat{\rho}} &= \frac{ x \hat{\mathbf x} + y \hat{\mathbf y}}{\sqrt{x^2+y^2}} \\ \boldsymbol{\hat{\phi}} &= \frac{- y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2+y^2}} \\ \mathbf{\hat{z}} &= \mathbf{\hat{z}} \end{align}$	$\begin{align} \hat{\mathbf x} &= \cos\phi\boldsymbol{\hat{\rho}} - \sin\phi\boldsymbol{\hat{\phi}} \\ \hat{\mathbf y} &= \sin\phi\boldsymbol{\hat{\rho}} + \cos\phi\boldsymbol{\hat{\phi}} \\ \mathbf{\hat{z}} &= \mathbf{\hat{z}} \end{align}$	$\begin{align} \hat{\mathbf x} &= \sin\theta\cos\phi\boldsymbol{\hat{r}} + \cos\theta\cos\phi\boldsymbol{\hat{\theta}}-\sin\phi\boldsymbol{\hat{\phi}} \\ \hat{\mathbf y} &= \sin\theta\sin\phi\boldsymbol{\hat{r}} + \cos\theta\sin\phi\boldsymbol{\hat{\theta}}+\cos\phi\boldsymbol{\hat{\phi}} \\ \mathbf{\hat{z}} &= \cos\theta \boldsymbol{\hat{r}} - \sin\theta \boldsymbol{\hat{\theta}} \end{align}$	$\begin{align} \boldsymbol{\hat{\sigma}} &= \frac{\tau \hat{\mathbf x} - \sigma \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\ \boldsymbol{\hat{\tau}} &= \frac{\sigma \hat{\mathbf x} + \tau \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\ \mathbf{\hat{z}} &= \mathbf{\hat{z}} \end{align}$
Definition of unit vectors	$\begin{align} \mathbf{\hat{r}} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y} + z \mathbf{\hat{z}}}{\sqrt{x^2+y^2+z^2}} \\ \boldsymbol{\hat{\theta}} &= \frac{x z \hat{\mathbf x} + y z \hat{\mathbf y} - \left(x^2 + y^2\right) \mathbf{\hat{z}}}{\sqrt{x^2+y^2} \sqrt{x^2+y^2+z^2}} \\ \boldsymbol{\hat{\phi}} &= \frac{- y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2+y^2}} \end{align}$	$\begin{align} \mathbf{\hat{r}} &= \frac{\rho \boldsymbol{\hat{\rho}} + z \mathbf{\hat{z}}}{\sqrt{\rho^2 +z^2}} \\ \boldsymbol{\hat{\theta}} &= \frac{ z \boldsymbol{\hat{\rho}} - \rho \mathbf{\hat{z}}}{\sqrt{\rho^2 +z^2}} \\ \boldsymbol{\hat{\phi}} &= \boldsymbol{\hat{\phi}} \end{align}$	$\begin{align} \boldsymbol{\hat{\rho}} &= \sin\theta \mathbf{\hat{r}} + \cos\theta \boldsymbol{\hat{\theta}} \\ \boldsymbol{\hat{\phi}} &= \boldsymbol{\hat{\phi}} \\ \mathbf{\hat{z}} &= \cos\theta \mathbf{\hat{r}} - \sin\theta \boldsymbol{\hat{\theta}} \end{align}$	$\begin{matrix} \end{matrix}$
A vector field $\mathbf A$	$A_x \hat{\mathbf x} + A_y \hat{\mathbf y} + A_z \mathbf{\hat{z}}$	$A_\rho \boldsymbol{\hat{\rho}} + A_\phi \boldsymbol{\hat{\phi}} + A_z \mathbf{\hat{z}}$	$A_r \boldsymbol{\hat{r}} + A_\theta \boldsymbol{\hat{\theta}} + A_\phi \boldsymbol{\hat{\phi}}$	$A_\sigma \boldsymbol{\hat{\sigma}} + A_\tau \boldsymbol{\hat{\tau}} + A_\phi \mathbf{\hat{z}}$
$f$ scalar field Gradient $\nabla f$	${\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y} + {\partial f \over \partial z}\mathbf{\hat{z}}$	${\partial f \over \partial \rho}\boldsymbol{\hat{\rho}} + {1 \over \rho}{\partial f \over \partial \phi}\boldsymbol{\hat{\phi}} + {\partial f \over \partial z}\mathbf{\hat{z}}$	${\partial f \over \partial r}\boldsymbol{\hat{r}} + {1 \over r}{\partial f \over \partial \theta}\boldsymbol{\hat{\theta}} + {1 \over r\sin\theta}{\partial f \over \partial \phi}\boldsymbol{\hat{\phi}}$	$\frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \sigma}\boldsymbol{\hat{\sigma}} + \frac{1}{\sqrt{\sigma^{2} + \tau^{2}}} {\partial f \over \partial \tau}\boldsymbol{\hat{\tau}} + {\partial f \over \partial z}\mathbf{\hat{z}}$
Divergence $\nabla \cdot \mathbf{A}$	${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$	${1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho} + {1 \over \rho}{\partial A_\phi \over \partial \phi} + {\partial A_z \over \partial z}$	${1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right) + {1 \over r\sin\theta}{\partial A_\phi \over \partial \phi}$	$\frac{1}{\sigma^{2} + \tau^{2}}\left({\partial (\sqrt{\sigma^2+\tau^2} A_\sigma) \over \partial \sigma} + {\partial (\sqrt{\sigma^2+\tau^2} A_\tau) \over \partial \tau}\right) + {\partial A_z \over \partial z}$
Curl $\nabla \times \mathbf{A}$	$\begin{align} \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) &\hat{\mathbf x} + \\ + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) &\hat{\mathbf y} + \\ + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) &\mathbf{\hat{z}} \end{align}$	$\begin{align} \left( \frac{1}{\rho} \frac{\partial A_z}{\partial \phi} - \frac{\partial A_\phi}{\partial z} \right) &\boldsymbol{\hat{\rho}} \\ + \left( \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) &\boldsymbol{\hat{\phi}} \\ + \frac{1}{\rho} \left( \frac{\partial \left(\rho A_\phi\right)}{\partial \rho} - \frac{\partial A_\rho}{\partial \phi} \right) &\mathbf{\hat{z}} \end{align}$	$\begin{align} \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left(A_\phi\sin\theta \right) - \frac{\partial A_\theta}{\partial \phi} \right) &\boldsymbol{\hat{r}} \\ + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \phi} - \frac{\partial}{\partial r} \left( r A_\phi \right) \right) &\boldsymbol{\hat{\theta}} \\ + \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_\theta \right) - \frac{\partial A_r}{\partial \theta} \right) &\boldsymbol{\hat{\phi}} \end{align}$	$\begin{align} \left( \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \tau} - \frac{\partial A_\tau}{\partial z} \right) &\boldsymbol{\hat{\sigma}} \\ - \left( \frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \sigma} - \frac{\partial A_\sigma}{\partial z} \right) &\boldsymbol{\hat{\tau}} \\ + \frac{1}{\sqrt{\sigma^2 + \tau^2}} \left( \frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\sigma \right)}{\partial \tau} - \frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\tau \right)}{\partial \sigma} \right) &\mathbf{\hat{z}} \end{align}$
Laplace operator $\Delta f \equiv \nabla^2 f$	${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$	${1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right) + {1 \over \rho^2}{\partial^2 f \over \partial \phi^2} + {\partial^2 f \over \partial z^2}$	${1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial f \over \partial r}\right) \!+\!{1 \over r^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial f \over \partial \theta}\right) \!+\!{1 \over r^2\!\sin^2\theta}{\partial^2 f \over \partial \phi^2}$	$\frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} f}{\partial \sigma^{2}} + \frac{\partial^{2} f}{\partial \tau^{2}} \right) + \frac{\partial^{2} f}{\partial z^{2}}$
Vector Laplacian $\Delta \mathbf{A} \equiv \nabla^2 \mathbf{A}$	$\Delta A_x \hat{\mathbf x} + \Delta A_y \hat{\mathbf y} + \Delta A_z \mathbf{\hat{z}}$	— View by clicking [show] — $\begin{align} \mathopen{}\left(\Delta A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\phi}{\partial \phi}\right)\mathclose{} &\boldsymbol{\hat{\rho}} \\ + \mathopen{}\left(\Delta A_\phi - \frac{A_\phi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \phi}\right)\mathclose{} &\boldsymbol{\hat{\phi}} \\ + \Delta A_z &\mathbf{\hat{z}} \end{align}$	— View by clicking [show] — $\begin{align} \left(\Delta A_r - \frac{2 A_r}{r^2} - \frac{2}{r^2\sin\theta} \frac{\partial \left(A_\theta \sin\theta\right)}{\partial\theta} - \frac{2}{r^2\sin\theta}{\frac{\partial A_\phi}{\partial \phi}}\right) &\boldsymbol{\hat{r}} \\ + \left(\Delta A_\theta - \frac{A_\theta}{r^2\sin^2\theta} + \frac{2}{r^2} \frac{\partial A_r}{\partial \theta} - \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\phi}{\partial \phi}\right) &\boldsymbol{\hat{\theta}} \\ + \left(\Delta A_\phi - \frac{A_\phi}{r^2\sin^2\theta} + \frac{2}{r^2\sin\theta} \frac{\partial A_r}{\partial \phi} + \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\theta}{\partial \phi}\right) &\boldsymbol{\hat{\phi}} \end{align}$
Material derivative^[1] $(\mathbf{A} \cdot \nabla) \mathbf{B}$	$\mathbf{A} \cdot \nabla B_x \hat{\mathbf x} + \mathbf{A} \cdot \nabla B_y \hat{\mathbf y} + \mathbf{A} \cdot \nabla B_z \hat{\mathbf{z}}$	— View by clicking [show] — $\begin{align} \left(A_\rho \frac{\partial B_\rho}{\partial \rho}+\frac{A_\phi}{\rho}\frac{\partial B_\rho}{\partial \phi}+A_z\frac{\partial B_\rho}{\partial z}-\frac{A_\phi B_\phi}{\rho}\right) &\boldsymbol{\hat{\rho}} \\ + \left(A_\rho \frac{\partial B_\phi}{\partial \rho} + \frac{A_\phi}{\rho}\frac{\partial B_\phi}{\partial \phi} + A_z\frac{\partial B_\phi}{\partial z} + \frac{A_\phi B_\rho}{\rho}\right) &\boldsymbol{\hat{\phi}}\\ + \left(A_\rho \frac{\partial B_z}{\partial \rho}+\frac{A_\phi}{\rho}\frac{\partial B_z}{\partial \phi}+A_z\frac{\partial B_z}{\partial z}\right) &\mathbf{\hat{z}} \end{align}$	— View by clicking [show] — $\begin{align} \left( A_r \frac{\partial B_r}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_r}{\partial \theta} + \frac{A_\phi}{r\sin\theta} \frac{\partial B_r}{\partial \phi} - \frac{A_\theta B_\theta + A_\phi B_\phi}{r} \right) &\boldsymbol{\hat{r}} \\ + \left( A_r \frac{\partial B_\theta}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\theta}{\partial \theta} + \frac{A_\phi}{r\sin\theta} \frac{\partial B_\theta}{\partial \phi} + \frac{A_\theta B_r}{r} - \frac{A_\phi B_\phi\cot\theta}{r} \right) &\boldsymbol{\hat{\theta}} \\ + \left( A_r \frac{\partial B_\phi}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\phi}{\partial \theta} + \frac{A_\phi}{r\sin\theta} \frac{\partial B_\phi}{\partial \phi} + \frac{A_\phi B_r}{r} + \frac{A_\phi B_\theta \cot\theta}{r} \right) &\boldsymbol{\hat{\phi}} \end{align}$
Differential displacement	$d\mathbf{l} = dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \mathbf{\hat{z}}$	$d\mathbf{l} = d\rho \, \boldsymbol{\hat{\rho}} + \rho \, d\phi \, \boldsymbol{\hat{\phi}} + dz \, \mathbf{\hat{z}}$	$d\mathbf{l} = dr \, \mathbf{\hat{r}} + r \, d\theta \, \boldsymbol{\hat{\theta}} + r \, \sin\theta \, d\phi \, \boldsymbol{\hat{\phi}}$	$d\mathbf{l} = \sqrt{\sigma^2 + \tau^2} \, d\sigma \, \boldsymbol{\hat{\sigma}} + \sqrt{\sigma^2 + \tau^2} \, d\tau \, \boldsymbol{\hat{\tau}} + dz \, \mathbf{\hat{z}}$
Differential normal area $d \mathbf S$	$\begin{align} dy \, dz &\hat{\mathbf x} \\ + dx \, dz &\hat{\mathbf y} \\ + dx \, dy &\mathbf{\hat{z}} \end{align}$	$\begin{align} \rho \, d\phi \, dz &\boldsymbol{\hat{\rho}} \\ + d\rho \, dz &\boldsymbol{\hat{\phi}} \\ + \rho \, d\rho \, d\phi &\mathbf{\hat{z}} \end{align}$	$\begin{align} r^2 \sin\theta \, d\theta \, d\phi &\mathbf{\hat{r}} \\ + r \sin\theta \, dr \, d\phi &\boldsymbol{\hat{\theta}} \\ + r \, dr \, d\theta &\boldsymbol{\hat{\phi}} \end{align}$	$\begin{align} \sqrt{\sigma^2 + \tau^2} \, d\tau \, dz &\boldsymbol{\hat{\sigma}} \\ + \sqrt{\sigma^2 + \tau^2} \, d\sigma \, dz &\boldsymbol{\hat{\tau}} \\ + \left(\sigma^2 + \tau^2\right) \, d\sigma \, d\tau &\mathbf{\hat{z}} \end{align}$
Differential volume $dV$	$dx \, dy \, dz$	$\rho \, d\rho \, d\phi \, dz$	$r^2 \sin\theta \, dr \, d\theta \, d\phi$	$\left(\sigma^2 + \tau^2\right) d\sigma \, d\tau \, dz$
Non-trivial calculation rules: $\operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f = \nabla^2 f \equiv \Delta f$ $\operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0$ $\operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0$ $\operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ (Lagrange's formula for del) $\Delta (f g) = f \Delta g + 2 \nabla f \cdot \nabla g + g \Delta f$