Del in cylindrical and spherical coordinates

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This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Notes

• This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ):
  • The polar angle is denoted by θ: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
  • The azimuthal angle is denoted by φ: it is the angle between the x-axis and the projection of the radial vector onto the xy-plane.
• The function atan2(y, x) can be used instead of the mathematical function arctan(y/x) owing to its domain and image. The classical arctan function has an image of (−π/2, +π/2), whereas atan2 is defined to have an image of (−π, π].

Coordinate conversions

Conversion between Cartesian, cylindrical, and spherical coordinates

	Cartesian	Cylindrical	Spherical
Cartesian	$\begin{align} x &= x \\ y &= y \\ z &= z \end{align}$	$\begin{align} x &= \rho \cos\varphi \\ y &= \rho \sin\varphi \\ z &= z \end{align}$	$\begin{align} x &= r \sin\theta \cos\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\theta \end{align}$
Cylindrical	$\begin{align} \rho &= \sqrt{x^2 + y^2} \\ \varphi &= \arctan(y / x) \\ z &= z \end{align}$	$\begin{align} \rho &= \rho \\ \varphi &= \varphi \\ z &= z \end{align}$	$\begin{align} \rho &= r \sin\theta \\ \varphi &= \varphi \\ z &= r\cos\theta \end{align}$
Spherical	$\begin{align} r &= \sqrt{x^2+y^2+z^2} \\ \theta &= \arccos(z/r) \\ \varphi &= \arctan(y/x) \end{align}$	$\begin{align} r &= \sqrt{\rho^2 + z^2} \\ \theta &= \arctan{(\rho/z)} \\ \varphi &= \varphi \end{align}$	$\begin{align} r &= r \\ \theta &= \theta \\ \varphi &= \varphi \end{align}$

Unit vector conversions

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

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

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

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

Del formulae

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

Operation	Cartesian coordinates (x, y, z)	Cylindrical coordinates (ρ, φ, z)	Spherical coordinates (r, θ, φ)
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 ∇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 ∇ ⋅ 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 ∇ × A	$\begin{align} \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) &\hat{\mathbf x} \\ + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) &\hat{\mathbf y} \\ + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) &\hat{\mathbf z} \end{align}$	$\begin{align} \left( \frac{1}{\rho} \frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z} \right) &\hat{\boldsymbol \rho} \\ + \left( \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) &\hat{\boldsymbol \varphi} \\ + \frac{1}{\rho} \left( \frac{\partial \left(\rho A_\varphi\right)}{\partial \rho} - \frac{\partial A_\rho}{\partial \varphi} \right) &\hat{\mathbf z} \end{align}$	$\begin{align} \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left(A_\varphi\sin\theta \right) - \frac{\partial A_\theta}{\partial \varphi} \right) &\hat{\mathbf r} \\ + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \varphi} - \frac{\partial}{\partial r} \left( r A_\varphi \right) \right) &\hat{\boldsymbol \theta} \\ + \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_\theta \right) - \frac{\partial A_r}{\partial \theta} \right) &\hat{\boldsymbol \varphi} \end{align}$
Laplace operator ∇2f ≡ ∆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 ∇2A ≡ ∆A	$\nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z}$	— View by clicking [show] — $\begin{align} \mathopen{}\left(\nabla^2 A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\varphi}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \rho} \\ + \mathopen{}\left(\nabla^2 A_\varphi - \frac{A_\varphi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \varphi} \\ + \nabla^2 A_z &\hat{\mathbf z} \end{align}$	— View by clicking [show] — $\begin{align} \left(\nabla^2 A_r - \frac{2 A_r}{r^2} - \frac{2}{r^2\sin\theta} \frac{\partial \left(A_\theta \sin\theta\right)}{\partial\theta} - \frac{2}{r^2\sin\theta}{\frac{\partial A_\varphi}{\partial \varphi}}\right) &\hat{\mathbf r} \\ + \left(\nabla^2 A_\theta - \frac{A_\theta}{r^2\sin^2\theta} + \frac{2}{r^2} \frac{\partial A_r}{\partial \theta} - \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\varphi}{\partial \varphi}\right) &\hat{\boldsymbol \theta} \\ + \left(\nabla^2 A_\varphi - \frac{A_\varphi}{r^2\sin^2\theta} + \frac{2}{r^2\sin\theta} \frac{\partial A_r}{\partial \varphi} + \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\theta}{\partial \varphi}\right) &\hat{\boldsymbol \varphi} \end{align}$
Material derivative (A ⋅ ∇)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] — $\begin{align} \left( A_r \frac{\partial B_r}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_r}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_r}{\partial \varphi} - \frac{A_\theta B_\theta + A_\varphi B_\varphi}{r} \right) &\hat{\mathbf r} \\ + \left( A_r \frac{\partial B_\theta}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\theta}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\theta}{\partial \varphi} + \frac{A_\theta B_r}{r} - \frac{A_\varphi B_\varphi\cot\theta}{r} \right) &\hat{\boldsymbol \theta} \\ + \left( A_r \frac{\partial B_\varphi}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\varphi}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\varphi}{\partial \varphi} + \frac{A_\varphi B_r}{r} + \frac{A_\varphi B_\theta \cot\theta}{r} \right) &\hat{\boldsymbol \varphi} \end{align}$
Differential displacement dℓ	$dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}$	$d\rho \, \hat{\boldsymbol \rho} + \rho \, d\varphi \, \hat{\boldsymbol \varphi} + dz \, \hat{\mathbf z}$	$dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\varphi \, \hat{\boldsymbol \varphi}$
Differential normal area dS	$\begin{align} dy \, dz &\, \hat{\mathbf x} \\ {} + dx \, dz &\, \hat{\mathbf y} \\ {} + dx \, dy &\, \hat{\mathbf z} \end{align}$	$\begin{align} \rho \, d\varphi \, dz &\, \hat{\boldsymbol \rho} \\ {} + d\rho \, dz &\, \hat{\boldsymbol \varphi} \\ {} + \rho \, d\rho \, d\varphi &\, \hat{\mathbf z} \end{align}$	$\begin{align} r^2 \sin\theta \, d\theta \, d\varphi &\, \hat{\mathbf r} \\ {} + r \sin\theta \, dr \, d\varphi &\, \hat{\boldsymbol \theta} \\ {} + r \, dr \, d\theta &\, \hat{\boldsymbol \varphi} \end{align}$
Differential volume dV	$dx \, dy \, dz$	$\rho \, d\rho \, d\varphi \, dz$	$r^2 \sin\theta \, dr \, d\theta \, d\varphi$

Non-trivial calculation rules

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

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.