Del in cylindrical and spherical coordinates

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 \theta \in , \pi/math>: 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 \varphi \in , 2\pi/math>: it is the angle between the ''x''-axis and the projection of the radial vector onto the ''xy''-plane.
* The function can be used instead of the mathematical function owing to its domain and image. The classical arctan function has an image of , whereas atan2 is defined to have an image of .

Coordinate conversions

CAUTION: the operation $\arctan\left(\frac\right)$ must be interpreted as the two-argument inverse tangent, atan2.

Unit vector conversions

Del formula

{, class="wikitable" style="text-align: center;"
, + Table with the del operator in cartesian, cylindrical and spherical coordinates

, -
! Operation
! Cartesian coordinates
! Cylindrical coordinates
! Spherical coordinates ,
where is the polar angle and is the azimuthal angle

, -
! Vector field
, $A_x \hat{\mathbf x} + A_y \hat{\mathbf y} + A_z \hat{\mathbf z}$
, $A_\rho \hat{\boldsymbol \rho} + A_\varphi \hat{\boldsymbol \varphi} + A_z \hat{\mathbf z}$
, $A_r \hat{\mathbf r} + A_\theta \hat{\boldsymbol \theta} + A_\varphi \hat{\boldsymbol \varphi}$

, -
! Gradient
, ${\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y} + {\partial f \over \partial z}\hat{\mathbf z}$
, ${\partial f \over \partial \rho}\hat{\boldsymbol \rho} + {1 \over \rho}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi} + {\partial f \over \partial z}\hat{\mathbf z}$
, ${\partial f \over \partial r}\hat{\mathbf r} + {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta} + {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi}$

, -
! Divergence
, ${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$
, ${1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho} + {1 \over \rho}{\partial A_\varphi \over \partial \varphi} + {\partial A_z \over \partial z}$
, ${1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right) + {1 \over r\sin\theta}{\partial A_\varphi \over \partial \varphi}$

, -
! Curl
, $\begin{align} \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) &\hat{\mathbf x} \\ + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) &\hat{\mathbf y} \\ + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) &\hat{\mathbf z} \end{align}$
, $\begin{align} \left( \frac{1}{\rho}\frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z} \right) &\hat{\boldsymbol \rho} \\ + \left( \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) &\hat{\boldsymbol \varphi} \\ + \frac{1}{\rho} \left( \frac{\partial \left(\rho A_\varphi\right)}{\partial \rho} - \frac{\partial A_\rho}{\partial \varphi} \right) &\hat{\mathbf z} \end{align}$
, $\begin{align} \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left(A_\varphi\sin\theta \right) - \frac{\partial A_\theta}{\partial \varphi} \right) &\hat{\mathbf r} \\ {}+ \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \varphi} - \frac{\partial}{\partial r} \left( r A_\varphi \right) \right) &\hat{\boldsymbol \theta} \\ {}+ \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_{\theta} \right) - \frac{\partial A_r}{\partial \theta} \right) &\hat{\boldsymbol \varphi} \end{align}$

, -
! Laplace operator
, ${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$
, ${1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right) + {1 \over \rho^2}{\partial^2 f \over \partial \varphi^2} + {\partial^2 f \over \partial z^2}$
, ${1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial f \over \partial r}\right) \!+\!{1 \over r^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial f \over \partial \theta}\right) \!+\!{1 \over r^2\!\sin^2\theta}{\partial^2 f \over \partial \varphi^2}$

, -
! Vector Gradient
, $\begin{align}{}&\frac{\partial A_x}{\partial x} \hat{\mathbf x} \otimes \hat{\mathbf x} + \frac{\partial A_x}{\partial y} \hat{\mathbf x} \otimes \hat{\mathbf y} + \frac{\partial A_x}{\partial z} \hat{\mathbf x} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_y}{\partial x} \hat{\mathbf y} \otimes \hat{\mathbf x} + \frac{\partial A_y}{\partial y} \hat{\mathbf y} \otimes \hat{\mathbf y} + \frac{\partial A_y}{\partial z} \hat{\mathbf y} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_z}{\partial x} \hat{\mathbf z} \otimes \hat{\mathbf x} + \frac{\partial A_z}{\partial y} \hat{\mathbf z} \otimes \hat{\mathbf y} + \frac{\partial A_z}{\partial z} \hat{\mathbf z} \otimes \hat{\mathbf z}\end{align}$
, $\begin{align}{}&\frac{\partial A_\rho}{\partial \rho} \hat{\boldsymbol \rho} \otimes \hat{\boldsymbol \rho} + \left(\frac{1}{\rho}\frac{\partial A_\rho}{\partial \varphi}-\frac{A_\varphi}{\rho}\right) \hat{\boldsymbol \rho} \otimes \hat{\boldsymbol \varphi} + \frac{\partial A_\rho}{\partial z} \hat{\boldsymbol \rho} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_\varphi}{\partial \rho} \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \rho} + \left(\frac{1}{\rho}\frac{\partial A_\varphi}{\partial \varphi}+\frac{A_\rho}{\rho}\right) \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \varphi} + \frac{\partial A_\varphi}{\partial z} \hat{\boldsymbol \varphi} \otimes \hat{\mathbf z} \\ {}+ &\frac{\partial A_z}{\partial \rho} \hat{\mathbf z} \otimes \hat{\boldsymbol \rho} + \frac{1}{\rho}\frac{\partial A_z}{\partial \varphi} \hat{\mathbf z} \otimes \hat{\boldsymbol \varphi} + \frac{\partial A_z}{\partial z} \hat{\mathbf z} \otimes \hat{\mathbf z}\end{align}$
, $\begin{align}{}&\frac{\partial A_r}{\partial r} \hat{\mathbf r} \otimes \hat{\mathbf r} + \left(\frac{1}{r}\frac{\partial A_r}{\partial \theta}-\frac{A_\theta}{r}\right) \hat{\mathbf r} \otimes \hat{\boldsymbol \theta} + \left(\frac{1}{r \sin\theta}\frac{\partial A_r}{\partial \varphi} - \frac{A_\varphi}{r}\right) \hat{\mathbf r} \otimes \hat{\boldsymbol \varphi} \\ {}+ &\frac{\partial A_\theta}{\partial r} \hat{\boldsymbol \theta} \otimes \hat{\mathbf r} + \left(\frac{1}{r}\frac{\partial A_\theta}{\partial \theta}+\frac{A_r}{r}\right) \hat{\boldsymbol \theta} \otimes \hat{\boldsymbol \theta} + \left(\frac{1}{r \sin\theta}\frac{\partial A_\theta}{\partial \varphi} - \cot\theta \frac{A_\varphi}{r}\right) \hat{\boldsymbol \theta} \otimes \hat{\boldsymbol \varphi} \\ {}+ &\frac{\partial A_\varphi}{\partial r} \hat{\boldsymbol \varphi} \otimes \hat{\mathbf r} + \frac{1}{r}\frac{\partial A_\varphi}{\partial \theta} \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \theta} + \left(\frac{1}{r \sin\theta}\frac{\partial A_\varphi}{\partial \varphi} + \cot\theta \frac{A_\theta}{r} + \frac{A_r}{r}\right) \hat{\boldsymbol \varphi} \otimes \hat{\boldsymbol \varphi}\end{align}$

, -
! Vector Laplacian
, $\nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z}$
,
$\begin{align} \mathopen{}\left(\nabla^2 A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\varphi}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \rho} \\ + \mathopen{}\left(\nabla^2 A_\varphi - \frac{A_\varphi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \varphi} \\ {}+ \nabla^2 A_z &\hat{\mathbf z} \end{align}$

,
$\begin{align} \left(\nabla^2 A_r - \frac{2 A_r}{r^2} - \frac{2}{r^2\sin\theta} \frac{\partial \left(A_\theta \sin\theta\right)}{\partial\theta} - \frac{2}{r^2\sin\theta}{\frac{\partial A_\varphi}{\partial \varphi\right) &\hat{\mathbf r} \\ + \left(\nabla^2 A_\theta - \frac{A_\theta}{r^2\sin^2\theta} + \frac{2}{r^2} \frac{\partial A_r}{\partial \theta} - \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\varphi}{\partial \varphi}\right) &\hat{\boldsymbol \theta} \\ + \left(\nabla^2 A_\varphi - \frac{A_\varphi}{r^2\sin^2\theta} + \frac{2}{r^2\sin\theta} \frac{\partial A_r}{\partial \varphi} + \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\theta}{\partial \varphi}\right) &\hat{\boldsymbol \varphi} \end{align}$

, -
! Material derivative

, $\mathbf{A} \cdot \nabla B_x \hat{\mathbf x} + \mathbf{A} \cdot \nabla B_y \hat{\mathbf y} + \mathbf{A} \cdot \nabla B_z \hat{\mathbf{z$

, $\begin{align} \left(A_\rho \frac{\partial B_\rho}{\partial \rho}+\frac{A_\varphi}{\rho}\frac{\partial B_\rho}{\partial \varphi}+A_z\frac{\partial B_\rho}{\partial z}-\frac{A_\varphi B_\varphi}{\rho}\right) &\hat{\boldsymbol \rho} \\ + \left(A_\rho \frac{\partial B_\varphi}{\partial \rho} + \frac{A_\varphi}{\rho}\frac{\partial B_\varphi}{\partial \varphi} + A_z\frac{\partial B_\varphi}{\partial z} + \frac{A_\varphi B_\rho}{\rho}\right) &\hat{\boldsymbol \varphi}\\ + \left(A_\rho \frac{\partial B_z}{\partial \rho}+\frac{A_\varphi}{\rho}\frac{\partial B_z}{\partial \varphi}+A_z\frac{\partial B_z}{\partial z}\right) &\hat{\mathbf z} \end{align}$

,
$\begin{align} \left( A_r \frac{\partial B_r}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_r}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_r}{\partial \varphi} - \frac{A_\theta B_\theta + A_\varphi B_\varphi}{r} \right) &\hat{\mathbf r} \\ + \left( A_r \frac{\partial B_\theta}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\theta}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\theta}{\partial \varphi} + \frac{A_\theta B_r}{r} - \frac{A_\varphi B_\varphi\cot\theta}{r} \right) &\hat{\boldsymbol \theta} \\ + \left( A_r \frac{\partial B_\varphi}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\varphi}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\varphi}{\partial \varphi} + \frac{A_\varphi B_r}{r} + \frac{A_\varphi B_\theta \cot\theta}{r} \right) &\hat{\boldsymbol \varphi} \end{align}$

, -
! Tensor (not to be confused with 2nd order tensor divergence)

,
$\begin{align} \left(\frac{\partial T_{xx{\partial x}+\frac{\partial T_{yx{\partial y}+\frac{\partial T_{zx{\partial z}\right)&\hat{\mathbf x} \\ +\left(\frac{\partial T_{xy{\partial x}+\frac{\partial T_{yy{\partial y}+\frac{\partial T_{zy{\partial z}\right)&\hat{\mathbf y} \\ +\left(\frac{\partial T_{xz{\partial x}+\frac{\partial T_{yz{\partial y}+\frac{\partial T_{zz{\partial z}\right)&\hat{\mathbf z} \end{align}$

,
\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\varphi} \\
+\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}

,
$\begin{align} \left[\frac{\partial T_{rr{\partial r}+2\frac{T_{rrr+\frac1r\frac{\partial T_{\theta r{\partial\theta}+\frac{\cot\theta}rT_{\theta r}+\frac1{r\sin\theta}\frac{\partial T_{\varphi r{\partial\varphi}-\frac1r(T_{\theta\theta}+T_{\varphi\varphi})\right]&\hat{\mathbf r} \\ +\left[\frac{\partial T_{r\theta{\partial r}+2\frac{T_{r\thetar+\frac1r\frac{\partial T_{\theta\theta{\partial\theta}+\frac{\cot\theta}rT_{\theta\theta}+\frac1{r\sin\theta}\frac{\partial T_{\varphi\theta{\partial\varphi}+\frac{T_{\theta rr-\frac{\cot\theta}rT_{\varphi\varphi}\right]&\hat{\boldsymbol\theta} \\ +\left[\frac{\partial T_{r\varphi{\partial r}+2\frac{T_{r\varphir+\frac1r\frac{\partial T_{\theta\varphi{\partial\theta}+\frac1{r\sin\theta}\frac{\partial T_{\varphi\varphi{\partial\varphi}+\frac {T_{\varphi r{r}+\frac{\cot\theta}{r} (T_{\theta\varphi}+T_{\varphi\theta})\right]&\hat{\boldsymbol\varphi} \end{align}$

, -
! Differential displacement
, $dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}$
, $d\rho \, \hat{\boldsymbol \rho} + \rho \, d\varphi \, \hat{\boldsymbol \varphi} + dz \, \hat{\mathbf z}$
, $dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\varphi \, \hat{\boldsymbol \varphi}$

, -
! Differential normal area
, $\begin{align} dy \, dz &\, \hat{\mathbf x} \\ {} + dx \, dz &\, \hat{\mathbf y} \\ {} + dx \, dy &\, \hat{\mathbf z} \end{align}$
, $\begin{align} \rho \, d\varphi \, dz &\, \hat{\boldsymbol \rho} \\ {} + d\rho \, dz &\, \hat{\boldsymbol \varphi} \\ {} + \rho \, d\rho \, d\varphi &\, \hat{\mathbf z} \end{align}$
, $\begin{align} r^2 \sin\theta \, d\theta \, d\varphi &\, \hat{\mathbf r} \\ {} + r \sin\theta \, dr \, d\varphi &\, \hat{\boldsymbol \theta} \\ {} + r \, dr \, d\theta &\, \hat{\boldsymbol \varphi} \end{align}$

, -
! Differential volume
, $dx \, dy \, dz$
, $\rho \, d\rho \, d\varphi \, dz$
, $r^2 \sin\theta \, dr \, d\theta \, d\varphi$

: This page uses $\theta$ for the polar angle and $\varphi$ for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses $\theta$ for the azimuthal angle and $\varphi$ for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch $\theta$ and $\varphi$ in the formulae shown in the table above.

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)
# $\nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f$

Cartesian derivation

$\begin{align} \operatorname{div} \mathbf A = \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S{\iiint_V dV} &= \frac{A_x(x+dx)\,dy\,dz - A_x(x)\,dy\,dz + A_y(y+dy)\,dx\,dz - A_y(y)\,dx\,dz + A_z(z+dz)\,dx\,dy - A_z(z)\,dx\,dy}{dx\,dy\,dz} \\ &= \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \end{align}$

$\begin{align} (\operatorname{curl} \mathbf A)_x = \lim_{S^{\perp \mathbf{\hat x\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} &= \frac{A_z(y+dy)\,dz - A_z(y)\,dz + A_y(z)\,dy - A_y(z+dz)\,dy }{dy\,dz} \\ &= \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \end{align}$

The expressions for $(\operatorname{curl} \mathbf A)_y$ and $(\operatorname{curl} \mathbf A)_z$ are found in the same way.

Cylindrical derivation

$\begin{align} \operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S{\iiint_V dV} \\ &= \frac{A_\rho(\rho+d\rho)(\rho+d\rho)\,d\phi\, dz - A_\rho(\rho)\rho \,d\phi \,dz + A_\phi(\phi+d\phi)\,d\rho\, dz - A_\phi(\phi)\,d\rho\, dz + A_z(z+dz)\,d\rho\, (\rho +d\rho/2)\,d\phi - A_z(z)\,d\rho (\rho +d\rho/2)\, d\phi}{\rho \,d\phi \,d\rho\, dz} \\ &= \frac 1 \rho \frac{\partial (\rho A_\rho)}{\partial \rho} + \frac 1 \rho \frac{\partial A_\phi}{\partial \phi} + \frac{\partial A_z}{\partial z} \end{align}$

$\begin{align} (\operatorname{curl} \mathbf A)_\rho &= \lim_{S^{\perp \boldsymbol{\hat \rho\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} \\ &= \frac{A_\phi (z)(\rho+d\rho)\,d\phi - A_\phi(z+dz)(\rho+d\rho)\,d\phi + A_z(\phi + d\phi)\,dz - A_z(\phi)\,dz}{(\rho+d\rho)\,d\phi \,dz} \\ &= -\frac{\partial A_\phi}{\partial z} + \frac{1}{\rho} \frac{\partial A_z}{\partial \phi} \end{align}$

$\begin{align} (\operatorname{curl} \mathbf A)_\phi &= \lim_{S^{\perp \boldsymbol{\hat \phi\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} \\ &= \frac{A_z (\rho)\,dz - A_z(\rho + d\rho)\,dz + A_\rho(z+dz)\,d\rho - A_\rho(z)\,d\rho}{d\rho \,dz} \\ &= -\frac{\partial A_z}{\partial \rho} + \frac{\partial A_\rho}{\partial z} \end{align}$

$\begin{align} (\operatorname{curl} \mathbf A)_z &= \lim_{S^{\perp \boldsymbol{\hat z\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} \\ &= \frac{A_\rho(\phi)\,d\rho - A_\rho(\phi + d\phi)\,d\rho + A_\phi(\rho + d\rho)(\rho + d\rho)\,d\phi - A_\phi(\rho)\rho \,d\phi}{\rho \,d\rho \,d\phi} \\ &= -\frac{1}{\rho}\frac{\partial A_\rho}{\partial \phi} + \frac{1}{\rho} \frac{\partial (\rho A_\phi)}{\partial \rho} \end{align}$

$\begin{align} \operatorname{curl} \mathbf A &= (\operatorname{curl} \mathbf A)_\rho \hat{\boldsymbol \rho} + (\operatorname{curl} \mathbf A)_\phi \hat{\boldsymbol \phi} + (\operatorname{curl} \mathbf A)_z \hat{\boldsymbol z} \\ &= \left(\frac{1}{\rho} \frac{\partial A_z}{\partial \phi} -\frac{\partial A_\phi}{\partial z} \right) \hat{\boldsymbol \rho} + \left(\frac{\partial A_\rho}{\partial z}-\frac{\partial A_z}{\partial \rho} \right) \hat{\boldsymbol \phi} + \frac{1}{\rho}\left(\frac{\partial (\rho A_\phi)}{\partial \rho} - \frac{\partial A_\rho}{\partial \phi} \right) \hat{\boldsymbol z} \end{align}$

Spherical derivation

$\begin{align} \operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S{\iiint_V dV} \\ &= \frac{A_r(r+dr)(r+dr)\,d\theta\, (r+dr)\sin\theta \,d\phi - A_r(r)r\,d\theta\, r\sin\theta \,d\phi + A_\theta(\theta+d\theta)\sin(\theta + d\theta)r\, dr\, d\phi - A_\theta(\theta)\sin(\theta)r \,dr \,d\phi + A_\phi(\phi + d\phi)r\,dr\, d\theta - A_\phi(\phi)r\,dr \,d\theta}{dr\,r\,d\theta\,r\sin\theta\, d\phi} \\ &= \frac{1}{r^2}\frac{\partial (r^2A_r)}{\partial r} + \frac{1}{r \sin\theta} \frac{\partial(A_\theta\sin\theta)}{\partial \theta} + \frac{1}{r \sin\theta} \frac{\partial A_\phi}{\partial \phi} \end{align}$

$\begin{align} (\operatorname{curl} \mathbf A)_r = \lim_{S^{\perp \boldsymbol{\hat r\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} &= \frac{A_\theta(\phi)r \,d\theta + A_\phi(\theta + d\theta)r \sin(\theta + d\theta)\, d\phi - A_\theta(\phi + d\phi)r \,d\theta - A_\phi(\theta)r\sin(\theta)\, d\phi}{r\, d\theta\,r\sin\theta \,d\phi} \\ &= \frac{1}{r\sin\theta}\frac{\partial(A_\phi \sin\theta)}{\partial \theta} - \frac{1}{r\sin\theta} \frac{\partial A_\theta}{\partial \phi} \end{align}$

$\begin{align} (\operatorname{curl} \mathbf A)_\theta = \lim_{S^{\perp \boldsymbol{\hat \theta\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} &= \frac{A_\phi(r)r \sin\theta \,d\phi + A_r(\phi + d\phi)\,dr - A_\phi(r+dr)(r+dr)\sin\theta \,d\phi - A_r(\phi)\,dr}{dr \, r \sin \theta \,d\phi} \\ &= \frac{1}{r\sin\theta}\frac{\partial A_r}{\partial \phi} - \frac{1}{r} \frac{\partial (rA_\phi)}{\partial r} \end{align}$

$\begin{align} (\operatorname{curl} \mathbf A)_\phi = \lim_{S^{\perp \boldsymbol{\hat \phi\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell{\iint_{S} dS} &= \frac{A_r(\theta)\,dr + A_\theta(r+dr)(r+dr)\,d\theta - A_r(\theta+d\theta)\,dr - A_\theta(r) r \,d\theta}{r\,dr\, d\theta} \\ &= \frac{1}{r}\frac{\partial(rA_\theta)}{\partial r} - \frac{1}{r} \frac{\partial A_r}{\partial \theta} \end{align}$

$\operatorname{curl} \mathbf A = (\operatorname{curl} \mathbf A)_r \, \hat{\boldsymbol r} + (\operatorname{curl} \mathbf A)_\theta \, \hat{\boldsymbol \theta} + (\operatorname{curl} \mathbf A)_\phi \, \hat{\boldsymbol \phi} = \frac{1}{r\sin\theta} \left(\frac{\partial(A_\phi \sin\theta)}{\partial \theta}-\frac{\partial A_\theta}{\partial \phi} \right) \hat{\boldsymbol r} +\frac{1}{r} \left(\frac{1}{\sin\theta}\frac{\partial A_r}{\partial \phi} - \frac{\partial (rA_\phi)}{\partial r} \right) \hat{\boldsymbol \theta} + \frac{1}{r}\left(\frac{\partial(rA_\theta)}{\partial r} - \frac{\partial A_r}{\partial \theta} \right) \hat{\boldsymbol \phi}$

Unit vector conversion formula

The unit vector of a coordinate parameter ''u'' is defined in such a way that a small positive change in ''u'' causes the position vector $\mathbf r$ to change in $\mathbf u$ direction.

Therefore,
$\frac{\partial {\mathbf r{\partial u} = \frac{\partial{s{\partial u} \mathbf u$
where is the arc length parameter.

For two sets of coordinate systems $u_i$ and $v_j$, according to chain rule,
$d\mathbf r = \sum_{i} \frac{\partial \mathbf r}{\partial u_i} \, du_i = \sum_{i} \frac{\partial s}{\partial u_i} \hat{\mathbf u}_i du_i = \sum_{j} \frac{\partial s}{\partial v_j} \hat{\mathbf v}_j \, dv_j = \sum_{j}\frac{\partial s}{\partial v_j} \hat{\mathbf v}_j \sum_{i} \frac{\partial v_j}{\partial u_i} \, du_i = \sum_{i} \sum_{j} \frac{\partial s}{\partial v_j} \frac{\partial v_j}{\partial u_i} \hat{\mathbf v}_j \, du_i.$

Now, we isolate the $i$^th component. For $i{\neq}k$, let $\mathrm d u_k=0$. Then divide on both sides by $\mathrm d u_i$ to get:
$\frac{\partial s}{\partial u_i} \hat{\mathbf u}_i = \sum_{j} \frac{\partial s}{\partial v_j} \frac{\partial v_j}{\partial u_i} \hat{\mathbf v}_j.$

See also

* Del
* Orthogonal coordinates
* Curvilinear coordinates
* Vector fields in cylindrical and spherical coordinates

References

{{Reflist

External links

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

Vector calculus
Coordinate systems