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
		From
		Cartesian	Cylindrical	Spherical
To	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	${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\varphi &=\arctan \left({\frac {y}{x}}\right)\\z&=z\end{aligned}}$	N/A	${\begin{aligned}\rho &=r\sin \theta \\\varphi &=\varphi \\z&=r\cos \theta \end{aligned}}$
	Spherical	${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos \left({\frac {z}{r}}\right)\\\varphi &=\arctan \left({\frac {y}{x}}\right)\end{aligned}}$	${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {\left({\frac {\rho }{z}}\right)}\\\varphi &=\varphi \end{aligned}}$	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	${\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$	N/A	${\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}$
Spherical	${\begin{aligned}{\hat {\mathbf {r} }}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {\left(x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}\right)z-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt {x^{2}+y^{2}+z^{2}}}{\sqrt {x^{2}+y^{2}}}}}\\{\hat {\boldsymbol {\varphi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}$	${\begin{aligned}{\hat {\mathbf {r} }}&={\frac {\rho {\hat {\boldsymbol {\rho }}}+z{\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {z{\hat {\boldsymbol {\rho }}}-\rho {\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\varphi }}}&={\hat {\boldsymbol {\varphi }}}\end{aligned}}$	N/A

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

Del formula

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 $\nabla \times A$	$\begin{align} \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) &\hat{\mathbf x} \\ + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) &\hat{\mathbf y} \\ + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) &\hat{\mathbf z} \end{align}$	${\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right)&{\hat {\boldsymbol {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\hat {\boldsymbol {\varphi }}}\\{}+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\varphi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \varphi }}\right)&{\hat {\mathbf {z} }}\end{aligned}}$	${\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\varphi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \varphi }}\right)&{\hat {\mathbf {r} }}\\{}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {\partial }{\partial r}}\left(rA_{\varphi }\right)\right)&{\hat {\boldsymbol {\theta }}}\\{}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\hat {\boldsymbol {\varphi }}}\end{aligned}}$
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] — ${\begin{aligned}{\mathopen {}}\left(\nabla ^{2}A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\varphi }}{\partial \varphi }}\right){\mathclose {}}&{\hat {\boldsymbol {\rho }}}\\+{\mathopen {}}\left(\nabla ^{2}A_{\varphi }-{\frac {A_{\varphi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\mathclose {}}&{\hat {\boldsymbol {\varphi }}}\\{}+\nabla ^{2}A_{z}&{\hat {\mathbf {z} }}\end{aligned}}$	— 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^α^[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] — ${\begin{aligned}\left({\frac {\partial T_{xx}}{\partial x}}+{\frac {\partial T_{yx}}{\partial y}}+{\frac {\partial T_{zx}}{\partial z}}\right)&{\hat {\mathbf {x} }}\\+\left({\frac {\partial T_{xy}}{\partial x}}+{\frac {\partial T_{yy}}{\partial y}}+{\frac {\partial T_{zy}}{\partial z}}\right)&{\hat {\mathbf {y} }}\\+\left({\frac {\partial T_{xz}}{\partial x}}+{\frac {\partial T_{yz}}{\partial y}}+{\frac {\partial T_{zz}}{\partial z}}\right)&{\hat {\mathbf {z} }}\end{aligned}}$	— View by clicking [show] — ${\begin{aligned}\left[{\frac {\partial T_{\rho \rho }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi \rho }}{\partial \varphi }}+{\frac {\partial T_{z\rho }}{\partial z}}+{\frac {1}{\rho }}(T_{\rho \rho }-T_{\varphi \varphi })\right]&{\hat {\boldsymbol {\rho }}}\\+\left[{\frac {\partial T_{\rho \varphi }}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {\partial T_{z\varphi }}{\partial z}}+{\frac {1}{\rho }}(T_{\rho \varphi }+T_{\varphi \rho })\right]&{\hat {\boldsymbol {\phi }}}\\+\left[{\frac {\partial T_{\rho z}}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial T_{\varphi z}}{\partial \varphi }}+{\frac {\partial T_{zz}}{\partial z}}+{\frac {T_{\rho z}}{\rho }}\right]&{\hat {\mathbf {z} }}\end{aligned}}$	— View by clicking [show] — ${\begin{aligned}\left[{\frac {\partial T_{rr}}{\partial r}}+2{\frac {T_{rr}}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta r}}{\partial \theta }}+{\frac {\cot \theta }{r}}T_{\theta r}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi r}}{\partial \varphi }}-{\frac {1}{r}}(T_{\theta \theta }+T_{\varphi \varphi })\right]&{\hat {\mathbf {r} }}\\+\left[{\frac {\partial T_{r\theta }}{\partial r}}+2{\frac {T_{r\theta }}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta \theta }}{\partial \theta }}+{\frac {\cot \theta }{r}}T_{\theta \theta }+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi \theta }}{\partial \varphi }}+{\frac {T_{\theta r}}{r}}-{\frac {\cot \theta }{r}}T_{\varphi \varphi }\right]&{\hat {\boldsymbol {\theta }}}\\+\left[{\frac {\partial T_{r\varphi }}{\partial r}}+2{\frac {T_{r\varphi }}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta \varphi }}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {T_{\varphi r}}{r}}+{\frac {\cot \theta }{r}}(T_{\theta \varphi }+T_{\varphi \theta })\right]&{\hat {\boldsymbol {\varphi }}}\end{aligned}}$
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 $dV$	$dx \, dy \, dz$	$\rho \, d\rho \, d\varphi \, dz$	$r^2 \sin\theta \, dr \, d\theta \, d\varphi$

^α 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. This page uses

\theta

for the polar angle and

\varphi

for the azimuthal angle, which is common notation in physics. In order to get the mathematics formulae, switch

\theta

and

\varphi

in the formulae shown in the table above.

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

Cartesian derivation

${\begin{aligned}\operatorname {div} \mathbf {A} =\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}&={\frac {A_{x}(x+dx)dydz-A_{x}(x)dydz+A_{y}(y+dy)dxdz-A_{y}(y)dxdz+A_{z}(z+dz)dxdy-A_{z}(z)dxdy}{dxdydz}}\\&={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}$

${\begin{aligned}(\operatorname {curl} \mathbf {A} )_{x}=\lim _{S^{\perp \mathbf {\hat {x}} }\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{z}(y+dy)dz-A_{z}(y)dz+A_{y}(z)dy-A_{y}(z+dz)dy}{dydz}}\\&={\frac {\partial A_{y}}{\partial z}}-{\frac {\partial A_{z}}{\partial y}}\end{aligned}}$

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

Cylindrical derivation

${\begin{aligned}\operatorname {div} \mathbf {A} =\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}&={\frac {A_{\rho }(\rho +d\rho )(\rho +d\rho )d\phi \,dz-A_{\rho }(\rho )\rho d\phi \,dz+A_{\phi }(\phi +d\phi )d\rho \,dz-A_{\phi }(\phi )d\rho dz+A_{z}(z+dz)d\rho \,\rho d\phi -A_{z}(z)d\rho \,\rho d\phi }{d\rho \,\rho d\phi \,dz}}\\&={\frac {1}{\rho }}{\frac {\partial (\rho A_{\rho })}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}$

$(\operatorname {curl} \mathbf {A} )_{\rho }=\lim _{S^{\perp {\boldsymbol {\hat {\rho }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}$

Spherical derivation

${\begin{aligned}\operatorname {div} \mathbf {A} =\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}&={\frac {A_{r}(r+dr)(r+dr)d\theta \,(r+dr)\sin \theta d\phi -A_{r}(r)rd\theta \,r\sin \theta d\phi +...}{dr\,rd\theta \,r\sin \theta d\phi }}\\&={\frac {1}{r^{2}}}{\frac {\partial (r^{2}A_{r})}{\partial r}}+...\end{aligned}}$

${\begin{aligned}(\operatorname {curl} \mathbf {A} )_{r}=\lim _{S^{\perp {\boldsymbol {\hat {r}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\theta }(\phi )\,rd\theta +A_{\phi }(\theta +d\theta )\,r\sin(\theta +d\theta )d\phi -A_{\theta }(\phi +d\phi )\,rd\theta -A_{\phi }(\theta )\,r\sin \theta d\phi }{rd\theta \,r\sin \theta d\phi }}\\&={\frac {1}{r\sin \theta }}{\frac {\partial (\sin(\theta )A_{\phi })}{\partial \theta }}-{\frac {1}{r\sin \theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\end{aligned}}$

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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