Del in cylindrical and spherical coordinates

Short description: Mathematical gradient operator in certain coordinate systems

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 [math]\displaystyle{ \theta \in [0, \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 [math]\displaystyle{ \varphi \in [0, 2\pi] }[/math]: 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	[math]\displaystyle{ \begin{align} x&=x\\ y&=y\\ z&=z\\ \end{align} }[/math]	[math]\displaystyle{ \begin{align} x &= \rho \cos\varphi \\ y &= \rho \sin\varphi \\ z &= z \end{align} }[/math]	[math]\displaystyle{ \begin{align} x &= r \sin\theta \cos\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\theta \\ \end{align} }[/math]
	Cylindrical	[math]\displaystyle{ \begin{align} \rho &= \sqrt{x^2 + y^2} \\ \varphi &= \arctan\left(\frac{y}{x}\right) \\ z &= z \end{align} }[/math]	[math]\displaystyle{ \begin{align} \rho &=\rho\\ \varphi &=\varphi\\ z&=z\\ \end{align} }[/math]	[math]\displaystyle{ \begin{align} \rho &= r \sin\theta \\ \varphi &= \varphi \\ z &= r\cos\theta \end{align} }[/math]
	Spherical	[math]\displaystyle{ \begin{align} r &= \sqrt{x^2 + y^2 + z^2} \\ \theta &= \arctan\left(\frac{\sqrt{x^2 + y^2}}{z}\right) \\ \varphi &= \arctan\left(\frac{y}{x}\right) \end{align} }[/math]	[math]\displaystyle{ \begin{align} r &= \sqrt{\rho^2 + z^2} \\ \theta &= \arctan{\left(\frac{\rho}{z}\right)} \\ \varphi &= \varphi \end{align} }[/math]	[math]\displaystyle{ \begin{align} r&=r\\\theta &=\theta \\\varphi &=\varphi \end{align} }[/math]

CAUTION: the operation [math]\displaystyle{ \arctan\left(\frac{A}{B}\right) }[/math] 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	[math]\displaystyle{ \begin{align} x&=x\\ y&=y\\ z&=z\\ \end{align} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]
Cylindrical	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \begin{align} \rho &=\rho\\ \varphi &=\varphi\\ z&=z\\ \end{align} }[/math]	[math]\displaystyle{ \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} }[/math]
Spherical	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \begin{align} r&=r\\\theta &=\theta \\\varphi &=\varphi \end{align} }[/math]

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

	Cartesian	Cylindrical	Spherical
Cartesian	[math]\displaystyle{ \begin{align} x&=x\\ y&=y\\ z&=z\\ \end{align} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]
Cylindrical	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \begin{align} \rho &=\rho\\ \varphi &=\varphi\\ z&=z\\ \end{align} }[/math]	[math]\displaystyle{ \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} }[/math]
Spherical	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \begin{align} r&=r\\\theta &=\theta \\\varphi &=\varphi \end{align} }[/math]

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 the azimuthal angle
Vector field A	[math]\displaystyle{ A_x \hat{\mathbf x} + A_y \hat{\mathbf y} + A_z \hat{\mathbf z} }[/math]	[math]\displaystyle{ A_\rho \hat{\boldsymbol \rho} + A_\varphi \hat{\boldsymbol \varphi} + A_z \hat{\mathbf z} }[/math]	[math]\displaystyle{ A_r \hat{\mathbf r} + A_\theta \hat{\boldsymbol \theta} + A_\varphi \hat{\boldsymbol \varphi} }[/math]
Gradient ∇f	[math]\displaystyle{ {\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y} + {\partial f \over \partial z}\hat{\mathbf z} }[/math]	[math]\displaystyle{ {\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} }[/math]	[math]\displaystyle{ {\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} }[/math]
Divergence ∇ ⋅ A	[math]\displaystyle{ {\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z} }[/math]	[math]\displaystyle{ {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} }[/math]	[math]\displaystyle{ {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} }[/math]
Curl ∇ × A	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]
Laplace operator ∇2f ≡ ∆f	[math]\displaystyle{ {\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2} }[/math]	[math]\displaystyle{ {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} }[/math]	[math]\displaystyle{ {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} }[/math]
Vector gradient ∇A	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]
Vector Laplacian ∇2A ≡ ∆A	[math]\displaystyle{ \nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]
Directional derivative (A ⋅ ∇)B	[math]\displaystyle{ \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}} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]
Tensor divergence ∇ ⋅ T	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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+\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} }[/math]
Differential displacement dℓ	[math]\displaystyle{ dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z} }[/math]	[math]\displaystyle{ d\rho \, \hat{\boldsymbol \rho} + \rho \, d\varphi \, \hat{\boldsymbol \varphi} + dz \, \hat{\mathbf z} }[/math]	[math]\displaystyle{ dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\varphi \, \hat{\boldsymbol \varphi} }[/math]
Differential normal area dS	[math]\displaystyle{ \begin{align} dy \, dz &\, \hat{\mathbf x} \\ {} + dx \, dz &\, \hat{\mathbf y} \\ {} + dx \, dy &\, \hat{\mathbf z} \end{align} }[/math]	[math]\displaystyle{ \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} }[/math]	[math]\displaystyle{ \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} }[/math]
Differential volume dV	[math]\displaystyle{ dx \, dy \, dz }[/math]	[math]\displaystyle{ \rho \, d\rho \, d\varphi \, dz }[/math]	[math]\displaystyle{ r^2 \sin\theta \, dr \, d\theta \, d\varphi }[/math]

^α This page uses [math]\displaystyle{ \theta }[/math] for the polar angle and [math]\displaystyle{ \varphi }[/math] for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses [math]\displaystyle{ \theta }[/math] for the azimuthal angle and [math]\displaystyle{ \varphi }[/math] for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch [math]\displaystyle{ \theta }[/math] and [math]\displaystyle{ \varphi }[/math] in the formulae shown in the table above.

Calculation rules

1. [math]\displaystyle{ \operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f \equiv \nabla^2 f }[/math]
2. [math]\displaystyle{ \operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0 }[/math]
3. [math]\displaystyle{ \operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0 }[/math]
4. [math]\displaystyle{ \operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} }[/math] (Lagrange's formula for del)
5. [math]\displaystyle{ \nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f }[/math]
6. [math]\displaystyle{ \nabla^{2}\left(\mathbf{P}\cdot\mathbf{Q}\right)=\mathbf{Q}\cdot\nabla^{2}\mathbf{P}-\mathbf{P}\cdot\nabla^{2}\mathbf{Q}+2\nabla\cdot\left[\left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf{P}\times\nabla\times\mathbf{Q}\right]\quad }[/math] (From )

Cartesian derivation

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

The expressions for [math]\displaystyle{ (\operatorname{curl} \mathbf A)_y }[/math] and [math]\displaystyle{ (\operatorname{curl} \mathbf A)_z }[/math] are found in the same way.

Cylindrical derivation

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

Spherical derivation

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

[math]\displaystyle{ \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} }[/math]

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 [math]\displaystyle{ \mathbf r }[/math] to change in [math]\displaystyle{ \mathbf u }[/math] direction.

Therefore, [math]\displaystyle{ \frac{\partial {\mathbf r}}{\partial u} = \frac{\partial{s}}{\partial u} \mathbf u }[/math] where s is the arc length parameter.

For two sets of coordinate systems [math]\displaystyle{ u_i }[/math] and [math]\displaystyle{ v_j }[/math], according to chain rule, [math]\displaystyle{ 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. }[/math]

Now, we isolate the [math]\displaystyle{ i }[/math]^th component. For [math]\displaystyle{ i{\neq}k }[/math], let [math]\displaystyle{ \mathrm d u_k=0 }[/math]. Then divide on both sides by [math]\displaystyle{ \mathrm d u_i }[/math] to get: [math]\displaystyle{ \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. }[/math]

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.