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].

Coordinate conversions

Conversion between Cartesian, cylindrical, and spherical coordinates
	Cartesian	Cylindrical	Spherical
Cartesian	N/A	$\begin{align} x &= \rho \cos\varphi \\ y &= \rho \sin\varphi \\ z &= z \end{align}$	$\begin{align} x &= r \sin\theta \cos\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\theta \end{align}$
Cylindrical	$\begin{align} \rho &= \sqrt{x^2 + y^2} \\ \varphi &= \arctan\left(\frac{y}{x}\right) \\ z &= z \end{align}$	N/A	$\begin{align} \rho &= r \sin\theta \\ \varphi &= \varphi \\ z &= r\cos\theta \end{align}$
Spherical	$\begin{align} r &= \sqrt{x^2 + y^2 + z^2} \\ \theta &= \arccos\left(\frac{z}{r}\right) \\ \varphi &= \arctan\left(\frac{y}{x}\right) \end{align}$	$\begin{align} r &= \sqrt{\rho^2 + z^2} \\ \theta &= \arctan{\left(\frac{\rho}{z}\right)} \\ \varphi &= \varphi \end{align}$	N/A

Unit vector conversions

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of *destination* coordinates
	Cartesian	Cylindrical	Spherical
Cartesian	N/A	$\begin{align} \hat{\mathbf x} &= \cos\varphi \hat{\boldsymbol \rho} - \sin\varphi \hat{\boldsymbol \varphi} \\ \hat{\mathbf y} &= \sin\varphi \hat{\boldsymbol \rho} + \cos\varphi \hat{\boldsymbol \varphi} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$	$\begin{align} \hat{\mathbf x} &= \sin\theta \cos\varphi \hat{\mathbf r} + \cos\theta \cos\varphi \hat{\boldsymbol \theta} - \sin\varphi \hat{\boldsymbol \varphi} \\ \hat{\mathbf y} &= \sin\theta \sin\varphi \hat{\mathbf r} + \cos\theta \sin\varphi \hat{\boldsymbol \theta} + \cos\varphi \hat{\boldsymbol \varphi} \\ \hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta} \end{align}$
Cylindrical	$\begin{align} \hat{\boldsymbol \rho} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\ \hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$	N/A	$\begin{align} \hat{\boldsymbol \rho} &= \sin\theta \hat{\mathbf r} + \cos\theta \hat{\boldsymbol \theta} \\ \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\ \hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta} \end{align}$
Spherical	$\begin{align} \hat{\mathbf r} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y} + z \hat{\mathbf z}}{\sqrt{x^2 + y^2 + z^2}} \\ \hat{\boldsymbol \theta} &= \frac{\left(x \hat{\mathbf x} + y \hat{\mathbf y}\right) z - \left(x^2 + y^2\right) \hat{\mathbf z}}{\sqrt{x^2 + y^2 + z^2} \sqrt{x^2 + y^2}} \\ \hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \end{align}$	$\begin{align} \hat{\mathbf r} &= \frac{\rho \hat{\boldsymbol \rho} + z \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\ \hat{\boldsymbol \theta} &= \frac{z \hat{\boldsymbol \rho} - \rho \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\ \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \end{align}$	N/A

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of *source* coordinates
	Cartesian	Cylindrical	Spherical
Cartesian	N/A	$\begin{align} \hat{\mathbf x} &= \frac{x \hat{\boldsymbol \rho} - y \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\ \hat{\mathbf y} &= \frac{y \hat{\boldsymbol \rho} + x \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$	$\begin{align} \hat{\mathbf x} &= \frac{x \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) - y \sqrt{x^2 + y^2 + z^2} \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2} \sqrt{x^2 + y^2 + z^2}} \\ \hat{\mathbf y} &= \frac{y \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) + x \sqrt{x^2 + y^2 + z^2} \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2} \sqrt{x^2 + y^2 + z^2}} \\ \hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \sqrt{x^2 + y^2} \hat{\boldsymbol \theta}}{\sqrt{x^2 + y^2 + z^2}} \end{align}$
Cylindrical	$\begin{align} \hat{\boldsymbol \rho} &= \cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y} \\ \hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$	N/A	$\begin{align} \hat{\boldsymbol \rho} &= \frac{\rho \hat{\mathbf r} + z \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}} \\ \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\ \hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \rho \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}} \end{align}$
Spherical	$\begin{align} \hat{\mathbf r} &= \sin\theta \left(\cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y}\right) + \cos\theta \hat{\mathbf z} \\ \hat{\boldsymbol \theta} &= \cos\theta \left(\cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y}\right) - \sin\theta \hat{\mathbf z} \\ \hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y} \end{align}$	$\begin{align} \hat{\mathbf r} &= \sin\theta \hat{\boldsymbol \rho} + \cos\theta \hat{\mathbf z} \\ \hat{\boldsymbol \theta} &= \cos\theta \hat{\boldsymbol \rho} - \sin\theta \hat{\mathbf z} \\ \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \end{align}$	N/A

Del formulae

Table with the del operator in cartesian, cylindrical and spherical coordinates
Operation	Cartesian coordinates $(x, y, z)$	Cylindrical coordinates $(ρ, φ, z)$	Spherical coordinates $(r, θ, φ)$ , where θ is the polar angle and φ is azimuthal
A vector field $A$	$A_x \hat{\mathbf x} + A_y \hat{\mathbf y} + A_z \hat{\mathbf z}$	$A_\rho \hat{\boldsymbol \rho} + A_\varphi \hat{\boldsymbol \varphi} + A_z \hat{\mathbf z}$	$A_r \hat{\mathbf r} + A_\theta \hat{\boldsymbol \theta} + A_\varphi \hat{\boldsymbol \varphi}$
Gradient $\nabla f$	${\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y} + {\partial f \over \partial z}\hat{\mathbf z}$	${\partial f \over \partial \rho}\hat{\boldsymbol \rho} + {1 \over \rho}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi} + {\partial f \over \partial z}\hat{\mathbf z}$	${\partial f \over \partial r}\hat{\mathbf r} + {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta} + {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi}$
Divergence $\nabla \cdot A$	${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$	${1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho} + {1 \over \rho}{\partial A_\varphi \over \partial \varphi} + {\partial A_z \over \partial z}$	${1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right) + {1 \over r\sin\theta}{\partial A_\varphi \over \partial \varphi}$
Curl $\nabla \times A$	$\begin{align} \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) &\hat{\mathbf x} \\ + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) &\hat{\mathbf y} \\ + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) &\hat{\mathbf z} \end{align}$	$\begin{align} \left( \frac{1}{\rho} \frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z} \right) &\hat{\boldsymbol \rho} \\ + \left( \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) &\hat{\boldsymbol \varphi} \\ {}+ \frac{1}{\rho} \left( \frac{\partial \left(\rho A_\varphi\right)}{\partial \rho} - \frac{\partial A_\rho}{\partial \varphi} \right) &\hat{\mathbf z} \end{align}$	$\begin{align} \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left(A_\varphi\sin\theta \right) - \frac{\partial A_\theta}{\partial \varphi} \right) &\hat{\mathbf r} \\ {}+ \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \varphi} - \frac{\partial}{\partial r} \left( r A_\varphi \right) \right) &\hat{\boldsymbol \theta} \\ {}+ \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_\theta \right) - \frac{\partial A_r}{\partial \theta} \right) &\hat{\boldsymbol \varphi} \end{align}$
Laplace operator $\nabla 2 f \equiv ∆ f$	${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$	${1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right) + {1 \over \rho^2}{\partial^2 f \over \partial \varphi^2} + {\partial^2 f \over \partial z^2}$	${1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial f \over \partial r}\right) \!+\!{1 \over r^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial f \over \partial \theta}\right) \!+\!{1 \over r^2\!\sin^2\theta}{\partial^2 f \over \partial \varphi^2}$
Vector Laplacian $\nabla 2 A \equiv ∆ A$	$\nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z}$	— View by clicking [show] — $\begin{align} \mathopen{}\left(\nabla^2 A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\varphi}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \rho} \\ + \mathopen{}\left(\nabla^2 A_\varphi - \frac{A_\varphi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \varphi} \\ {}+ \nabla^2 A_z &\hat{\mathbf z} \end{align}$	— View by clicking [show] — $\begin{align} \left(\nabla^2 A_r - \frac{2 A_r}{r^2} - \frac{2}{r^2\sin\theta} \frac{\partial \left(A_\theta \sin\theta\right)}{\partial\theta} - \frac{2}{r^2\sin\theta}{\frac{\partial A_\varphi}{\partial \varphi}}\right) &\hat{\mathbf r} \\ + \left(\nabla^2 A_\theta - \frac{A_\theta}{r^2\sin^2\theta} + \frac{2}{r^2} \frac{\partial A_r}{\partial \theta} - \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\varphi}{\partial \varphi}\right) &\hat{\boldsymbol \theta} \\ + \left(\nabla^2 A_\varphi - \frac{A_\varphi}{r^2\sin^2\theta} + \frac{2}{r^2\sin\theta} \frac{\partial A_r}{\partial \varphi} + \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\theta}{\partial \varphi}\right) &\hat{\boldsymbol \varphi} \end{align}$
Material derivative^[1] $(A \cdot \nabla) B$	$\mathbf{A} \cdot \nabla B_x \hat{\mathbf x} + \mathbf{A} \cdot \nabla B_y \hat{\mathbf y} + \mathbf{A} \cdot \nabla B_z \hat{\mathbf{z}}$	— View by clicking [show] — $\begin{align} \left(A_\rho \frac{\partial B_\rho}{\partial \rho}+\frac{A_\varphi}{\rho}\frac{\partial B_\rho}{\partial \varphi}+A_z\frac{\partial B_\rho}{\partial z}-\frac{A_\varphi B_\varphi}{\rho}\right) &\hat{\boldsymbol \rho} \\ + \left(A_\rho \frac{\partial B_\varphi}{\partial \rho} + \frac{A_\varphi}{\rho}\frac{\partial B_\varphi}{\partial \varphi} + A_z\frac{\partial B_\varphi}{\partial z} + \frac{A_\varphi B_\rho}{\rho}\right) &\hat{\boldsymbol \varphi}\\ + \left(A_\rho \frac{\partial B_z}{\partial \rho}+\frac{A_\varphi}{\rho}\frac{\partial B_z}{\partial \varphi}+A_z\frac{\partial B_z}{\partial z}\right) &\hat{\mathbf z} \end{align}$	— View by clicking [show] — $\begin{align} \left( A_r \frac{\partial B_r}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_r}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_r}{\partial \varphi} - \frac{A_\theta B_\theta + A_\varphi B_\varphi}{r} \right) &\hat{\mathbf r} \\ + \left( A_r \frac{\partial B_\theta}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\theta}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\theta}{\partial \varphi} + \frac{A_\theta B_r}{r} - \frac{A_\varphi B_\varphi\cot\theta}{r} \right) &\hat{\boldsymbol \theta} \\ + \left( A_r \frac{\partial B_\varphi}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\varphi}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\varphi}{\partial \varphi} + \frac{A_\varphi B_r}{r} + \frac{A_\varphi B_\theta \cot\theta}{r} \right) &\hat{\boldsymbol \varphi} \end{align}$
Tensor divergence $\nabla \cdot T$	— View by clicking [show] — $\begin{align} \left(\frac{\partial T_{xx}}{\partial x}+\frac{\partial T_{yx}}{\partial y}+\frac{\partial T_{zx}}{\partial z}\right)&\hat{\mathbf x} \\ +\left(\frac{\partial T_{xy}}{\partial x}+\frac{\partial T_{yy}}{\partial y}+\frac{\partial T_{zy}}{\partial z}\right)&\hat{\mathbf y} \\ +\left(\frac{\partial T_{xz}}{\partial x}+\frac{\partial T_{yz}}{\partial y}+\frac{\partial T_{zz}}{\partial z}\right)&\hat{\mathbf z} \end{align}$	— View by clicking [show] — $\begin{align} \left[\frac{\partial T_{\rho\rho}}{\partial\rho}+\frac1\rho\frac{\partial T_{\varphi\rho}}{\partial\varphi}+\frac{\partial T_{z\rho}}{\partial z}+\frac1\rho(T_{\rho\rho}-T_{\varphi\varphi})\right]&\hat{\boldsymbol\rho} \\ +\left[\frac{\partial T_{\rho\varphi}}{\partial\rho}+\frac1\rho\frac{\partial T_{\varphi\varphi}}{\partial\varphi}+\frac{\partial T_{z\varphi}}{\partial z}+\frac1\rho(T_{\rho\varphi}+T_{\varphi\rho})\right]&\hat{\boldsymbol\phi} \\ +\left[\frac{\partial T_{\rho z}}{\partial\rho}+\frac1\rho\frac{\partial T_{\varphi z}}{\partial\varphi}+\frac{\partial T_{zz}}{\partial z}+\frac{T_{\rho z}}\rho\right]&\hat{\mathbf z} \end{align}$	— View by clicking [show] — $\begin{align} \left[\frac{\partial T_{rr}}{\partial r}+2\frac{T_{rr}}r+\frac1r\frac{\partial T_{\theta r}}{\partial\theta}+\frac{\cot\theta}rT_{\theta r}+\frac1{r\sin\theta}\frac{\partial T_{\varphi r}}{\partial\varphi}-\frac1r(T_{\theta\theta}+T_{\varphi\varphi})\right]&\hat{\mathbf r} \\ +\left[\frac{\partial T_{r\theta}}{\partial r}+2\frac{T_{r\theta}}r+\frac1r\frac{\partial T_{\theta\theta}}{\partial\theta}+\frac{\cot\theta}rT_{\theta\theta}+\frac1{r\sin\theta}\frac{\partial T_{\varphi\theta}}{\partial\varphi}+\frac{T_{\theta r}}r-\frac{\cot\theta}rT_{\varphi\varphi}\right]&\hat{\boldsymbol\theta} \\ +\left[\frac{\partial T_{r\varphi}}{\partial r}+2\frac{T_{r\varphi}}r+\frac{\sin\theta}r\frac{\partial T_{\theta\varphi}}{\partial\theta}+\frac{\cos\theta}rT_{\theta\varphi}+\frac1{r\sin\theta}\frac{\partial T_{\varphi\varphi}}{\partial\varphi}+\frac1r(T_{\varphi r}+T_{\varphi\theta})\right]&\hat{\boldsymbol\varphi} \end{align}$
Differential displacement $d ℓ$	$dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}$	$d\rho \, \hat{\boldsymbol \rho} + \rho \, d\varphi \, \hat{\boldsymbol \varphi} + dz \, \hat{\mathbf z}$	$dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\varphi \, \hat{\boldsymbol \varphi}$
Differential normal area $d S$	$\begin{align} dy \, dz &\, \hat{\mathbf x} \\ {} + dx \, dz &\, \hat{\mathbf y} \\ {} + dx \, dy &\, \hat{\mathbf z} \end{align}$	$\begin{align} \rho \, d\varphi \, dz &\, \hat{\boldsymbol \rho} \\ {} + d\rho \, dz &\, \hat{\boldsymbol \varphi} \\ {} + \rho \, d\rho \, d\varphi &\, \hat{\mathbf z} \end{align}$	$\begin{align} r^2 \sin\theta \, d\theta \, d\varphi &\, \hat{\mathbf r} \\ {} + r \sin\theta \, dr \, d\varphi &\, \hat{\boldsymbol \theta} \\ {} + r \, dr \, d\theta &\, \hat{\boldsymbol \varphi} \end{align}$
Differential volume $dV$	$dx \, dy \, dz$	$\rho \, d\rho \, d\varphi \, dz$	$r^2 \sin\theta \, dr \, d\theta \, d\varphi$

Non-trivial calculation rules

$\operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f \equiv \nabla^2 f$
$\operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0$
$\operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0$
$\operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ (Lagrange's formula for del)
$\nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f$

References

↑ Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011.

External links

Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates.

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