Del in cylindrical and spherical coordinates explained

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 [ 0 , \pi] : 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 [ 0 , 2 \pi] : 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

! colspan="3"

From
Cartesian	Cylindrical	Spherical
To	Cartesian	\begin{align} x& = x\\ y& = y\\ z& = z\\ \end{align}	\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( (y)/(x) \right) \\ z & = z \end{align}	\begin{align} \rho & = \rho\\ \varphi & = \varphi\\ z& = z\\ \end{align}	\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 & = \arctan\left( (\sqrt{x ^{2} + y ^{2})/() }\right) \\ \varphi &= \arctan\left(\frac\right)\end	\begin{align} r & = \sqrt{\rho ^{2} + z ^{2}} \\ \theta & = \arctan{\left( (\rho)/(z) \right)} \\ \varphi & = \varphi \end{align}	\begin{align} r& = r\\\theta & = \theta \\\varphi & = \varphi \end{align}

Note that the operation \arctan\left( (A)/(B) \right) must be interpreted as the two-argument inverse tangent, atan2.

Unit vector conversions

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates

	Cartesian	Cylindrical	Spherical
Cartesian	\begin{align} \hat{ x}&=\hat{ x}\\ \hat{ y}&=\hat{ y}\\ \hat{ z}&=\hat{ z}\\ \end{align}	\begin{align} \hat{ x} & = \cos\varphi \hat{\boldsymbol \rho} - \sin\varphi \hat{\boldsymbol \varphi} \\ \hat{ y} & = \sin\varphi \hat{\boldsymbol \rho} + \cos\varphi \hat{\boldsymbol \varphi} \\ \hat{ z} & = \hat{ z} \end{align}	\begin{align} \hat{ x} & = \sin\theta \cos\varphi \hat{ r} + \cos\theta \cos\varphi \hat{\boldsymbol \theta} - \sin\varphi \hat{\boldsymbol \varphi} \\ \hat{ y} & = \sin\theta \sin\varphi \hat{ r} + \cos\theta \sin\varphi \hat{\boldsymbol \theta} + \cos\varphi \hat{\boldsymbol \varphi} \\ \hat{ z} & = \cos\theta \hat{ r} - \sin\theta \hat{\boldsymbol \theta} \end{align}
Cylindrical	\begin{align} \hat{\boldsymbol \rho} & = (x \hat{ x)/(+) y \hat{ y}}{\sqrt{x ^{2} + y ^{2}}} \\ \hat{\boldsymbol \varphi} & = (- y \hat{ x)/(+) x \hat{ y}}{\sqrt{x ^{2} + y ^{2}}} \\ \hat{ z} & = \hat{ z} \end{align}	\begin{align} \hat{\boldsymbol \rho}& = \hat{\boldsymbol \rho}\\ \hat{\boldsymbol \varphi}& = \hat{\boldsymbol \varphi}\\ \hat{ z}&=\hat{ z}\\ \end{align}	\begin{align} \hat{\boldsymbol \rho} & = \sin\theta \hat{ r} + \cos\theta \hat{\boldsymbol \theta} \\ \hat{\boldsymbol \varphi} & = \hat{\boldsymbol \varphi} \\ \hat{ z} & = \cos\theta \hat{ r} - \sin\theta \hat{\boldsymbol \theta} \end{align}
Spherical	\begin{align} \hat{ r} & = (x \hat{ x)/(+) y \hat{ y} + z \hat{ z}}{\sqrt{x ^{2} + y ^{2} + z ^{2}}} \\ \hat{\boldsymbol \theta} & = (\left(x \hat{ x)/(+) y \hat{ y}\right) z - \left(x ^{2} + y ^{2\right)} \hat{ z}}{\sqrt{x ^{2} + y ^{2} + z ^{2}} \sqrt{x ^{2} + y ^{2}}} \\ \hat{\boldsymbol \varphi} & = (- y \hat{ x)/(+) x \hat{ y}}{\sqrt{x ^{2} + y ^{2}} \end{align}}	\begin{align} \hat{ r} & = (\rho \hat{\boldsymbol \rho)/(+) z \hat{ z}}{\sqrt{\rho ^{2} + z ^{2}}} \\ \hat{\boldsymbol \theta} & = (z \hat{\boldsymbol \rho)/(-) \rho \hat{ z}}{\sqrt{\rho ^{2} + z ^{2}}} \\ \hat{\boldsymbol \varphi} & = \hat{\boldsymbol \varphi} \end{align}	\begin{align} \hat{ r}&=\hat{ r}\\ \hat{\boldsymbol \theta}&=\hat{\boldsymbol \theta}\\ \hat{\boldsymbol \varphi}& = \hat{\boldsymbol \varphi}\\ \end{align}

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of source coordinates

	Cartesian	Cylindrical	Spherical
Cartesian	\begin{align} \hat{ x}&=\hat{ x}\\ \hat{ y}&=\hat{ y}\\ \hat{ z}&=\hat{ z}\\ \end{align}	\begin{align} \hat{ x} & = (x \hat{\boldsymbol \rho)/(-) y \hat{\boldsymbol \varphi}}{\sqrt{x ^{2} + y ^{2}}} \\ \hat{ y} & = (y \hat{\boldsymbol \rho)/(+) x \hat{\boldsymbol \varphi}}{\sqrt{x ^{2} + y ^{2}}} \\ \hat{ z} & = \hat{ z} \end{align}	\begin{align} \hat{ x} & = (x \left(\sqrt{x ^{2} + y ^{2})/(\hat{ 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{ y} & = (y \left(\sqrt{x ^{2} + y ^{2})/(\hat{ 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{ z} & = (z \hat{ 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{ x} + \sin\varphi \hat{ y} \\ \hat{\boldsymbol \varphi} & = - \sin\varphi \hat{ x} + \cos\varphi \hat{ y} \\ \hat{ z} & = \hat{ z} \end{align}	\begin{align} \hat{\boldsymbol \rho}& = \hat{\boldsymbol \rho}\\ \hat{\boldsymbol \varphi}& = \hat{\boldsymbol \varphi}\\ \hat{ z}&=\hat{ z}\\ \end{align}	\begin{align} \hat{\boldsymbol \rho} & = (\rho \hat{ r)/(+) z \hat{\boldsymbol \theta}}{\sqrt{\rho ^{2} + z ^{2}}} \\ \hat{\boldsymbol \varphi} & = \hat{\boldsymbol \varphi} \\ \hat{ z} & = (z \hat{ r)/(-) \rho \hat{\boldsymbol \theta}}{\sqrt{\rho ^{2} + z ^{2}} \end{align}}
Spherical	\begin{align} \hat{ r} & = \sin\theta \left(\cos\varphi \hat{ x} + \sin\varphi \hat{ y}\right) + \cos\theta \hat{ z} \\ \hat{\boldsymbol \theta} & = \cos\theta \left(\cos\varphi \hat{ x} + \sin\varphi \hat{ y}\right) - \sin\theta \hat{ z} \\ \hat{\boldsymbol \varphi} & = - \sin\varphi \hat{ x} + \cos\varphi \hat{ y} \end{align}	\begin{align} \hat{ r} & = \sin\theta \hat{\boldsymbol \rho} + \cos\theta \hat{ z} \\ \hat{\boldsymbol \theta} & = \cos\theta \hat{\boldsymbol \rho} - \sin\theta \hat{ z} \\ \hat{\boldsymbol \varphi} & = \hat{\boldsymbol \varphi} \end{align}	\begin{align} \hat{ r}&=\hat{ r}\\ \hat{\boldsymbol \theta}&=\hat{\boldsymbol \theta}\\ \hat{\boldsymbol \varphi}& = \hat{\boldsymbol \varphi}\\ \end{align}

Del formula

Table with the del operator in cartesian, cylindrical and spherical coordinates

Operation			Spherical coordinates, where is the polar angle and is the azimuthal angle
Vector field	A _{x} \hat{ x} + A _{y} \hat{ y} + A _{z} \hat{ z}	A _{\rho} \hat{\boldsymbol \rho} + A _{\varphi} \hat{\boldsymbol \varphi} + A _{z} \hat{ z}	A _{r} \hat{ r} + A _{\theta} \hat{\boldsymbol \theta} + A _{\varphi} \hat{\boldsymbol \varphi}
Gradient	{\partial f \over \partial x}\hat{ x} + {\partial f \over \partial y}\hat{ y} + {\partial f \over \partial z}\hat{ 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{ z}	{\partial f \over \partial r}\hat{ 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( (\partial A _{z})/(\partial y) - (\partial A _{y})/(\partial z) \right) &\hat{ x} \\ + \left( (\partial A _{x})/(\partial z) - (\partial A _{z})/(\partial x) \right) &\hat{ y} \\ + \left( (\partial A _{y})/(\partial x) - (\partial A _{x})/(\partial y) \right) &\hat{ z} \end{align}	\begin{align} \left( (1)/(\rho) (\partial A _{z})/(\partial \varphi) - (\partial A _{\varphi})/(\partial z) \right) &\hat{\boldsymbol \rho} \\ + \left( (\partial A _{\rho})/(\partial z) - (\partial A _{z})/(\partial \rho) \right) &\hat{\boldsymbol \varphi} \\ + (1)/(\rho) \left( (\partial \left(\rho A _{\varphi\right)})/(\partial \rho) - (\partial A _{\rho})/(\partial \varphi) \right) &\hat{ z} \end{align}	\begin{align} (1)/(r\sin\theta) \left( (\partial)/(\partial \theta) \left(A _{\varphi\sin\theta} \right) - (\partial A _{\theta})/(\partial \varphi) \right) &\hat{ r} \\ {} + (1)/(r) \left( (1)/(\sin\theta) (\partial A _{r})/(\partial \varphi) - (\partial)/(\partial r) \left( r A _{\varphi} \right) \right) &\hat{\boldsymbol \theta} \\ {} + (1)/(r) \left( (\partial)/(\partial r) \left( r A _{\theta} \right) - (\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) \	+\!\	\left(\sin\theta \right)\!+\
Vector gradient	\begin{align}{}& (\partial A _{x})/(\partial x) \hat{ x} ⊗ \hat{ x} + (\partial A _{x})/(\partial y) \hat{ x} ⊗ \hat{ y} + (\partial A _{x})/(\partial z) \hat{ x} ⊗ \hat{ z} \ {} + & (\partial A _{y})/(\partial x) \hat{ y} ⊗ \hat{ x} + (\partial A _{y})/(\partial y) \hat{ y} ⊗ \hat{ y} + (\partial A _{y})/(\partial z) \hat{ y} ⊗ \hat{ z} \ {} + & (\partial A _{z})/(\partial x) \hat{ z} ⊗ \hat{ x} + (\partial A _{z})/(\partial y) \hat{ z} ⊗ \hat{ y} + (\partial A _{z})/(\partial z) \hat{ z} ⊗ \hat{ z}\end{align}	\begin{align}{}& (\partial A _{\rho})/(\partial \rho) \hat{\boldsymbol \rho} ⊗ \hat{\boldsymbol \rho} + \left( (1)/(\rho) (\partial A _{\rho})/(- \partial \varphi) (A _{\varphi})/(\rho) \right) \hat{\boldsymbol \rho} ⊗ \hat{\boldsymbol \varphi} + (\partial A _{\rho})/(\partial z) \hat{\boldsymbol \rho} ⊗ \hat{ z} \ {} + & (\partial A _{\varphi})/(\partial \rho) \hat{\boldsymbol \varphi} ⊗ \hat{\boldsymbol \rho} + \left( (1)/(\rho) (\partial A _{\varphi})/(+ \partial \varphi) (A _{\rho})/(\rho) \right) \hat{\boldsymbol \varphi} ⊗ \hat{\boldsymbol \varphi} + (\partial A _{\varphi})/(\partial z) \hat{\boldsymbol \varphi} ⊗ \hat{ z} \ {} + & (\partial A _{z})/(\partial \rho) \hat{ z} ⊗ \hat{\boldsymbol \rho} + (1)/(\rho) (\partial A _{z})/(\partial \varphi) \hat{ z} ⊗ \hat{\boldsymbol \varphi} + (\partial A _{z})/(\partial z) \hat{ z} ⊗ \hat{ z}\end{align}	\begin{align}{}& (\partial A _{r})/(\partial r) \hat{ r} ⊗ \hat{ r} + \left( (1)/(r) (\partial A _{r})/(- \partial \theta) (A _{\theta})/(r) \right) \hat{ r} ⊗ \hat{\boldsymbol \theta} + \left( (1)/(r \sin\theta) (\partial A _{r})/(\partial \varphi) - (A _{\varphi})/(r) \right) \hat{ r} ⊗ \hat{\boldsymbol \varphi} \ {} + & (\partial A _{\theta})/(\partial r) \hat{\boldsymbol \theta} ⊗ \hat{ r} + \left( (1)/(r) (\partial A _{\theta})/(+ \partial \theta) (A _{r})/(r) \right) \hat{\boldsymbol \theta} ⊗ \hat{\boldsymbol \theta} + \left( (1)/(r \sin\theta) (\partial A _{\theta})/(\partial \varphi) - \cot\theta (A _{\varphi})/(r) \right) \hat{\boldsymbol \theta} ⊗ \hat{\boldsymbol \varphi} \ {} + & (\partial A _{\varphi})/(\partial r) \hat{\boldsymbol \varphi} ⊗ \hat{ r} + (1)/(r) (\partial A _{\varphi})/(\partial \theta) \hat{\boldsymbol \varphi} ⊗ \hat{\boldsymbol \theta} + \left( (1)/(r \sin\theta) (\partial A _{\varphi})/(\partial \varphi) + \cot\theta (A _{\theta})/(r) + (A _{r})/(r) \right) \hat{\boldsymbol \varphi} ⊗ \hat{\boldsymbol \varphi}\end{align}
Vector Laplacian ^[1]	\nabla ^{2} A _{x} \hat{ x} + \nabla ^{2} A _{y} \hat{ y} + \nabla ^{2} A _{z} \hat{ z}	\begin{align} en{}\left(\nabla ^{2} A _{\rho} - (A _{\rho})/(\rho ^{2}) - (2)/(\rho ^{2}) (\partial A _{\varphi})/(\partial \varphi) \right) ose{} &\hat{\boldsymbol \rho} \\ + en{}\left(\nabla ^{2} A _{\varphi} - (A _{\varphi})/(\rho ^{2}) + (2)/(\rho ^{2}) (\partial A _{\rho})/(\partial \varphi) \right) ose{} &\hat{\boldsymbol \varphi} \\ {} + \nabla ^{2} A _{z} &\hat{ z} \end{align}	\begin{align} \left(\nabla ^{2} A _{r} - (2 A _{r})/(r ^{2}) - (2)/(r ^{2\sin\theta}) (\partial \left(A _{\theta} \sin\theta\right))/(\partial\theta) - (2)/({ r ^{2\sin\theta}) (\partial A _{\varphi})/(\partial \varphi) }\right) &\hat \\+ \left(\nabla^2 A_\theta - \frac + \frac \frac - \frac \frac\right) &\hat \\+ \left(\nabla^2 A_\varphi - \frac + \frac \frac + \frac \frac\right) &\hat\end
Directional derivative ^[2]	A ⋅ \nabla B _{x} \hat{ x} + A ⋅ \nabla B _{y} \hat{ y} + A ⋅ \nabla B _{z} \hat{ z }	\begin{align} \left(A _{\rho} (\partial B _{\rho})/(+ \partial \rho) (A _{\varphi})/(\rho) (\partial B _{\rho})/(\partial \varphi) + A _{z (\partial B _{\rho})/(\partial z)} - (A _{\varphi} B _{\varphi})/(\rho) \right) &\hat{\boldsymbol \rho} \\ + \left(A _{\rho} (\partial B _{\varphi})/(\partial \rho) + (A _{\varphi})/(\rho) (\partial B _{\varphi})/(\partial \varphi) + A _{z (\partial B _{\varphi})/(\partial z)} + (A _{\varphi} B _{\rho})/(\rho) \right) &\hat{\boldsymbol \varphi}\\ + \left(A _{\rho} (\partial B _{z})/(+ \partial \rho) (A _{\varphi})/(\rho) (\partial B _{z})/(\partial \varphi) + A _{z (\partial B _{z})/(\partial z)} \right) &\hat{ z} \end{align}	\begin{align} \left( A _{r} (\partial B _{r})/(\partial r) + (A _{\theta})/(r) (\partial B _{r})/(\partial \theta) + (A _{\varphi})/(r\sin\theta) (\partial B _{r})/(\partial \varphi) - (A _{\theta} B _{\theta} + A _{\varphi} B _{\varphi})/(r) \right) &\hat{ r} \\ + \left( A _{r} (\partial B _{\theta})/(\partial r) + (A _{\theta})/(r) (\partial B _{\theta})/(\partial \theta) + (A _{\varphi})/(r\sin\theta) (\partial B _{\theta})/(\partial \varphi) + (A _{\theta} B _{r})/(r) - (A _{\varphi} B _{\varphi\cot\theta})/(r) \right) &\hat{\boldsymbol \theta} \\ + \left( A _{r} (\partial B _{\varphi})/(\partial r) + (A _{\theta})/(r) (\partial B _{\varphi})/(\partial \theta) + (A _{\varphi})/(r\sin\theta) (\partial B _{\varphi})/(\partial \varphi) + (A _{\varphi} B _{r})/(r) + (A _{\varphi} B _{\theta} \cot\theta)/(r) \right) &\hat{\boldsymbol \varphi} \end{align}
Tensor divergence	\begin{align} \left( (\partial T _{xx})/(+ \partial x) (\partial T _{yx})/(+ \partial y) (\partial T _{zx})/(\right)&\hat{ x} \\ + \left( \partial z) (\partial T _{xy})/(+ \partial x) (\partial T _{yy})/(+ \partial y) (\partial T _{zy})/(\right)&\hat{ y} \\ + \left( \partial z) (\partial T _{xz})/(+ \partial x) (\partial T _{yz})/(+ \partial y) (\partial T _{zz})/(\partial z) \right)&\hat{ z} \end{align}	\begin{align} \left[ (\partial T _{\rho\rho})/(+ \partial\rho) (1 \rho (\partial T _{\varphi\rho})/(\partial\varphi))/(+ + (\partial T _{z\rho})/(\partial z)) (1 \rho(T)/(_{\rho\rho}) - T _{\varphi\varphi} )\right]&\hat{\boldsymbol\rho} \\ + \left[ (\partial T _{\rho\varphi})/(+ \partial\rho) (1 \rho (\partial T _{\varphi\varphi})/(\partial\varphi))/(+ + (\partial T _{z\varphi})/(\partial z)) (1 \rho(T)/(_{\rho\varphi}) + T _{\varphi\rho} )\right]&\hat{\boldsymbol\varphi} \\ + \left[ (\partial T _{\rho})/(+ \partial\rho) (1 \rho (\partial T _{\varphi})/(\partial\varphi))/(+ + (\partial T _{zz})/(\partial z)) (T _{\rho})/(\rho\right]&\hat{ z} \end{align})	\begin{align} \left[ (\partial T _{rr})/(+2 \partial r) (T _{rr})/(r + (1 r (\partial T _{\theta})/(\partial\theta))/(+ (\cot\theta)/(rT))) _{\theta} + (1 {r\sin\theta} (\partial T _{\varphi})/(\partial\varphi))/(- (1 r(T)/(_{\theta\theta})) + T _{\varphi\varphi} )\right]&\hat{r}\\ + \left[ (\partial T _{r\theta})/(+2 \partial r) (T _{r\theta})/(r + (1 r (\partial T _{\theta\theta})/(\partial\theta))/(+ (\cot\theta)/(rT))) _{\theta\theta} + (1 {r\sin\theta} (\partial T _{\varphi\theta})/(\partial\varphi))/(+ (T _{\theta})/(r - (\cot\theta)/(rT))) _{\varphi\varphi} \right]&\hat{\boldsymbol\theta} \\ + \left[ (\partial T _{r\varphi})/(+2 \partial r) (T _{r\varphi})/(+ r + (1 r (\partial T _{\theta\varphi})/(\partial\theta))/(+ (1 {r\sin\theta} (\partial T _{\varphi\varphi})/(\partial\varphi))/(+ (T _{\varphi})/(r)))) (\cot\theta)/(r) (T _{\theta\varphi} + T _{\varphi\theta} )\right]&\hat{\boldsymbol\varphi} \end{align}
Differential displacement	dx \hat{ x} + dy \hat{ y} + dz \hat{ z}	d\rho \hat{\boldsymbol \rho} + \rho d\varphi \hat{\boldsymbol \varphi} + dz \hat{ z}	dr \hat{ r} + r d\theta \hat{\boldsymbol \theta} + r \sin\theta d\varphi \hat{\boldsymbol \varphi}
Differential normal area	\begin{align} dy dz & \hat{ x} \\ {} + dx dz & \hat{ y} \\ {} + dx dy & \hat{ z} \end{align}	\begin{align} \rho d\varphi dz & \hat{\boldsymbol \rho} \\ {} + d\rho dz & \hat{\boldsymbol \varphi} \\ {} + \rho d\rho d\varphi & \hat{ z} \end{align}	\begin{align} r ^{2} \sin\theta d\theta d\varphi & \hat{ 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.

Defined in Cartesian coordinates as

\partial _{i} A ⊗ e _{i} . An alternative definition is e _{i} ⊗ \partial _{i} A .

Defined in Cartesian coordinates as

e _{i} ⋅ \partial _{i} T . An alternative definition is \partial _{i} T ⋅ e _{i} .

Calculation rules

\operatorname{div} \operatorname{grad} f \equiv \nabla ⋅ \nabla f \equiv \nabla ^{2} f \operatorname{curl} \operatorname{grad} f \equiv \nabla x \nabla f = 0 \operatorname{div} \operatorname{curl} A \equiv \nabla ⋅ (\nabla x A ) = 0 \operatorname{curl} \operatorname{curl} A \equiv \nabla x (\nabla x A ) = \nabla (\nabla ⋅ A ) - \nabla ^{2} A (Lagrange's formula for del) \nabla ^{2} (f g) = f \nabla ^{2} g + 2 \nabla f ⋅ \nabla g + g \nabla ^{2} f \nabla ^{2} \left( P ⋅ Q \right) = Q ⋅ \nabla ^{2} P - P ⋅ \nabla ^{2} Q +2 \nabla ⋅ \left[\left( P ⋅ \nabla\right) Q + P x \nabla x Q \right] (From ^[3])

Cartesian derivation

$$\backslash begin\backslash operatorname\; \backslash mathbf\; A\; =\; \backslash lim\_\; \backslash frac$$

&= \frac \\

&= \frac + \frac + \frac\end

The expressions for

(\operatorname{curl} A) _{y} and (\operatorname{curl} A) _{z} are found in the same way.

Cylindrical derivation

Spherical derivation

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

r to change in u direction.

Therefore, $$\backslash frac\; =\; \backslash frac\; \backslash mathbf\; u$$ where is the arc length parameter.

For two sets of coordinate systems

u _{i} and v _{j} , according to chain rule, $$d\backslash mathbf\; r\; =\; \backslash sum\_\; \backslash frac\; \backslash ,\; du\_i\; =\; \backslash sum\_\; \backslash frac\; \backslash hat\_i\; du\_i\; =\; \backslash sum\_\; \backslash frac\; \backslash hat\_j\; \backslash ,\; dv\_j\; =\; \backslash sum\_\backslash frac\; \backslash hat\_j\; \backslash sum\_\; \backslash frac\; \backslash ,\; du\_i\; =\; \backslash sum\_\; \backslash sum\_\; \backslash frac\; \backslash frac\; \backslash hat\_j\; \backslash ,\; du\_i.$$

Now, we isolate the

i ^th component. For i{ ≠ }k , let d u _{k=0} . Then divide on both sides by d u _{i} to get:$$\backslash frac\; \backslash hat\_i\; =\; \backslash sum\_\; \backslash frac\; \backslash frac\; \backslash hat\_j.$$

See also

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

External links

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

Notes and References

1. Book: Arfken . George . Weber . Hans . Harris . Frank . Mathematical Methods for Physicists . 2012 . Academic Press . 9789381269558 . 192 . Seventh . arfkenweber.
2. Web site: Convective Operator . Weisstein, Eric W. . Mathworld . 23 March 2011.
3. Fernández-Guasti . M. . Green's Second Identity for Vector Fields . ISRN Mathematical Physics . Hindawi Limited . 2012 . 2012 . 2090-4681 . 10.5402/2012/973968 . 1–7. free .