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 $θ \in [0, π]$ : 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 $φ \in [0, 2 π]$ : 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^[1]
		From
		Cartesian	Cylindrical	Spherical
To	Cartesian	$\begin{matrix} x & = x \\ y & = y \\ z & = z \end{matrix}$	$\begin{matrix} x & = ρ \cos φ \\ y & = ρ \sin φ \\ z & = z \end{matrix}$	$\begin{matrix} x & = r \sin θ \cos φ \\ y & = r \sin θ \sin φ \\ z & = r \cos θ \end{matrix}$
	Cylindrical	$\begin{matrix} ρ & = \sqrt{x^{2} + y^{2}} \\ φ & = \arctan (\frac{y}{x}) \\ z & = z \end{matrix}$	$\begin{matrix} ρ & = ρ \\ φ & = φ \\ z & = z \end{matrix}$	$\begin{matrix} ρ & = r \sin θ \\ φ & = φ \\ z & = r \cos θ \end{matrix}$
	Spherical	$\begin{matrix} r & = \sqrt{x^{2} + y^{2} + z^{2}} \\ θ & = \arccos (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}) \\ φ & = \arctan (\frac{y}{x}) \end{matrix}$	$\begin{matrix} r & = \sqrt{ρ^{2} + z^{2}} \\ θ & = \arctan (\frac{ρ}{z}) \\ φ & = φ \end{matrix}$	$\begin{matrix} r & = r \\ θ & = θ \\ φ & = φ \end{matrix}$

Note that the operation $\arctan (\frac{A}{B})$ 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^[1]
	Cartesian	Cylindrical	Spherical
Cartesian	$\begin{matrix} \hat{𝐱} & = \hat{𝐱} \\ \hat{𝐲} & = \hat{𝐲} \\ \hat{𝐳} & = \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝐱} & = \cos φ \hat{𝝆} - \sin φ \hat{𝝋} \\ \hat{𝐲} & = \sin φ \hat{𝝆} + \cos φ \hat{𝝋} \\ \hat{𝐳} & = \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝐱} & = \sin θ \cos φ \hat{𝐫} + \cos θ \cos φ \hat{𝜽} - \sin φ \hat{𝝋} \\ \hat{𝐲} & = \sin θ \sin φ \hat{𝐫} + \cos θ \sin φ \hat{𝜽} + \cos φ \hat{𝝋} \\ \hat{𝐳} & = \cos θ \hat{𝐫} - \sin θ \hat{𝜽} \end{matrix}$
Cylindrical	$\begin{matrix} \hat{𝝆} & = \frac{x \hat{𝐱} + y \hat{𝐲}}{\sqrt{x^{2} + y^{2}}} \\ \hat{𝝋} & = \frac{- y \hat{𝐱} + x \hat{𝐲}}{\sqrt{x^{2} + y^{2}}} \\ \hat{𝐳} & = \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝝆} & = \hat{𝝆} \\ \hat{𝝋} & = \hat{𝝋} \\ \hat{𝐳} & = \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝝆} & = \sin θ \hat{𝐫} + \cos θ \hat{𝜽} \\ \hat{𝝋} & = \hat{𝝋} \\ \hat{𝐳} & = \cos θ \hat{𝐫} - \sin θ \hat{𝜽} \end{matrix}$
Spherical	$\begin{matrix} \hat{𝐫} & = \frac{x \hat{𝐱} + y \hat{𝐲} + z \hat{𝐳}}{\sqrt{x^{2} + y^{2} + z^{2}}} \\ \hat{𝜽} & = \frac{(x \hat{𝐱} + y \hat{𝐲}) z - (x^{2} + y^{2}) \hat{𝐳}}{\sqrt{x^{2} + y^{2} + z^{2}} \sqrt{x^{2} + y^{2}}} \\ \hat{𝝋} & = \frac{- y \hat{𝐱} + x \hat{𝐲}}{\sqrt{x^{2} + y^{2}}} \end{matrix}$	$\begin{matrix} \hat{𝐫} & = \frac{ρ \hat{𝝆} + z \hat{𝐳}}{\sqrt{ρ^{2} + z^{2}}} \\ \hat{𝜽} & = \frac{z \hat{𝝆} - ρ \hat{𝐳}}{\sqrt{ρ^{2} + z^{2}}} \\ \hat{𝝋} & = \hat{𝝋} \end{matrix}$	$\begin{matrix} \hat{𝐫} & = \hat{𝐫} \\ \hat{𝜽} & = \hat{𝜽} \\ \hat{𝝋} & = \hat{𝝋} \end{matrix}$

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of *source* coordinates
	Cartesian	Cylindrical	Spherical
Cartesian	$\begin{matrix} \hat{𝐱} & = \hat{𝐱} \\ \hat{𝐲} & = \hat{𝐲} \\ \hat{𝐳} & = \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝐱} & = \frac{x \hat{𝝆} - y \hat{𝝋}}{\sqrt{x^{2} + y^{2}}} \\ \hat{𝐲} & = \frac{y \hat{𝝆} + x \hat{𝝋}}{\sqrt{x^{2} + y^{2}}} \\ \hat{𝐳} & = \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝐱} & = \frac{x (\sqrt{x^{2} + y^{2}} \hat{𝐫} + z \hat{𝜽}) - y \sqrt{x^{2} + y^{2} + z^{2}} \hat{𝝋}}{\sqrt{x^{2} + y^{2}} \sqrt{x^{2} + y^{2} + z^{2}}} \\ \hat{𝐲} & = \frac{y (\sqrt{x^{2} + y^{2}} \hat{𝐫} + z \hat{𝜽}) + x \sqrt{x^{2} + y^{2} + z^{2}} \hat{𝝋}}{\sqrt{x^{2} + y^{2}} \sqrt{x^{2} + y^{2} + z^{2}}} \\ \hat{𝐳} & = \frac{z \hat{𝐫} - \sqrt{x^{2} + y^{2}} \hat{𝜽}}{\sqrt{x^{2} + y^{2} + z^{2}}} \end{matrix}$
Cylindrical	$\begin{matrix} \hat{𝝆} & = \cos φ \hat{𝐱} + \sin φ \hat{𝐲} \\ \hat{𝝋} & = - \sin φ \hat{𝐱} + \cos φ \hat{𝐲} \\ \hat{𝐳} & = \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝝆} & = \hat{𝝆} \\ \hat{𝝋} & = \hat{𝝋} \\ \hat{𝐳} & = \hat{𝐳} \end{matrix}$	$\begin{matrix} \hat{𝝆} & = \frac{ρ \hat{𝐫} + z \hat{𝜽}}{\sqrt{ρ^{2} + z^{2}}} \\ \hat{𝝋} & = \hat{𝝋} \\ \hat{𝐳} & = \frac{z \hat{𝐫} - ρ \hat{𝜽}}{\sqrt{ρ^{2} + z^{2}}} \end{matrix}$
Spherical	$\begin{matrix} \hat{𝐫} & = \sin θ (\cos φ \hat{𝐱} + \sin φ \hat{𝐲}) + \cos θ \hat{𝐳} \\ \hat{𝜽} & = \cos θ (\cos φ \hat{𝐱} + \sin φ \hat{𝐲}) - \sin θ \hat{𝐳} \\ \hat{𝝋} & = - \sin φ \hat{𝐱} + \cos φ \hat{𝐲} \end{matrix}$	$\begin{matrix} \hat{𝐫} & = \sin θ \hat{𝝆} + \cos θ \hat{𝐳} \\ \hat{𝜽} & = \cos θ \hat{𝝆} - \sin θ \hat{𝐳} \\ \hat{𝝋} & = \hat{𝝋} \end{matrix}$	$\begin{matrix} \hat{𝐫} & = \hat{𝐫} \\ \hat{𝜽} & = \hat{𝜽} \\ \hat{𝝋} & = \hat{𝝋} \end{matrix}$

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^α
Vector field $A$	$A_{x} \hat{𝐱} + A_{y} \hat{𝐲} + A_{z} \hat{𝐳}$	$A_{ρ} \hat{𝝆} + A_{φ} \hat{𝝋} + A_{z} \hat{𝐳}$	$A_{r} \hat{𝐫} + A_{θ} \hat{𝜽} + A_{φ} \hat{𝝋}$
Gradient $\nabla f$ ^[1]	$\frac{\partial f}{\partial x} \hat{𝐱} + \frac{\partial f}{\partial y} \hat{𝐲} + \frac{\partial f}{\partial z} \hat{𝐳}$	$\frac{\partial f}{\partial ρ} \hat{𝝆} + \frac{1}{ρ} \frac{\partial f}{\partial φ} \hat{𝝋} + \frac{\partial f}{\partial z} \hat{𝐳}$	$\frac{\partial f}{\partial r} \hat{𝐫} + \frac{1}{r} \frac{\partial f}{\partial θ} \hat{𝜽} + \frac{1}{r \sin θ} \frac{\partial f}{\partial φ} \hat{𝝋}$
Divergence $\nabla \cdot A$ ^[1]	$\frac{\partial A_{x}}{\partial x} + \frac{\partial A_{y}}{\partial y} + \frac{\partial A_{z}}{\partial z}$	$\frac{1}{ρ} \frac{\partial (ρ A_{ρ})}{\partial ρ} + \frac{1}{ρ} \frac{\partial A_{φ}}{\partial φ} + \frac{\partial A_{z}}{\partial z}$	$\frac{1}{r^{2}} \frac{\partial (r^{2} A_{r})}{\partial r} + \frac{1}{r \sin θ} \frac{\partial}{\partial θ} (A_{θ} \sin θ) + \frac{1}{r \sin θ} \frac{\partial A_{φ}}{\partial φ}$
Curl $\nabla \times A$ ^[1]	$\begin{matrix} (\frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z}) & \hat{𝐱} \\ + (\frac{\partial A_{x}}{\partial z} - \frac{\partial A_{z}}{\partial x}) & \hat{𝐲} \\ + (\frac{\partial A_{y}}{\partial x} - \frac{\partial A_{x}}{\partial y}) & \hat{𝐳} \end{matrix}$	$\begin{matrix} (\frac{1}{ρ} \frac{\partial A_{z}}{\partial φ} - \frac{\partial A_{φ}}{\partial z}) & \hat{𝝆} \\ + (\frac{\partial A_{ρ}}{\partial z} - \frac{\partial A_{z}}{\partial ρ}) & \hat{𝝋} \\ + \frac{1}{ρ} (\frac{\partial (ρ A_{φ})}{\partial ρ} - \frac{\partial A_{ρ}}{\partial φ}) & \hat{𝐳} \end{matrix}$	$\begin{matrix} \frac{1}{r \sin θ} (\frac{\partial}{\partial θ} (A_{φ} \sin θ) - \frac{\partial A_{θ}}{\partial φ}) & \hat{𝐫} \\ + \frac{1}{r} (\frac{1}{\sin θ} \frac{\partial A_{r}}{\partial φ} - \frac{\partial}{\partial r} (r A_{φ})) & \hat{𝜽} \\ + \frac{1}{r} (\frac{\partial}{\partial r} (r A_{θ}) - \frac{\partial A_{r}}{\partial θ}) & \hat{𝝋} \end{matrix}$
Laplace operator $\nabla 2 f \equiv ∆ f$ ^[1]	$\frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}} + \frac{\partial^{2} f}{\partial z^{2}}$	$\frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \frac{\partial f}{\partial ρ}) + \frac{1}{ρ^{2}} \frac{\partial^{2} f}{\partial φ^{2}} + \frac{\partial^{2} f}{\partial z^{2}}$	$\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial f}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial f}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2} f}{\partial φ^{2}}$
Vector gradient $\nabla A$ ^β	$\begin{matrix} \frac{\partial A_{x}}{\partial x} \hat{𝐱} \otimes \hat{𝐱} + \frac{\partial A_{x}}{\partial y} \hat{𝐱} \otimes \hat{𝐲} + \frac{\partial A_{x}}{\partial z} \hat{𝐱} \otimes \hat{𝐳} \\ + & \frac{\partial A_{y}}{\partial x} \hat{𝐲} \otimes \hat{𝐱} + \frac{\partial A_{y}}{\partial y} \hat{𝐲} \otimes \hat{𝐲} + \frac{\partial A_{y}}{\partial z} \hat{𝐲} \otimes \hat{𝐳} \\ + & \frac{\partial A_{z}}{\partial x} \hat{𝐳} \otimes \hat{𝐱} + \frac{\partial A_{z}}{\partial y} \hat{𝐳} \otimes \hat{𝐲} + \frac{\partial A_{z}}{\partial z} \hat{𝐳} \otimes \hat{𝐳} \end{matrix}$	$\begin{matrix} \frac{\partial A_{ρ}}{\partial ρ} \hat{𝝆} \otimes \hat{𝝆} + (\frac{1}{ρ} \frac{\partial A_{ρ}}{\partial φ} - \frac{A_{φ}}{ρ}) \hat{𝝆} \otimes \hat{𝝋} + \frac{\partial A_{ρ}}{\partial z} \hat{𝝆} \otimes \hat{𝐳} \\ + & \frac{\partial A_{φ}}{\partial ρ} \hat{𝝋} \otimes \hat{𝝆} + (\frac{1}{ρ} \frac{\partial A_{φ}}{\partial φ} + \frac{A_{ρ}}{ρ}) \hat{𝝋} \otimes \hat{𝝋} + \frac{\partial A_{φ}}{\partial z} \hat{𝝋} \otimes \hat{𝐳} \\ + & \frac{\partial A_{z}}{\partial ρ} \hat{𝐳} \otimes \hat{𝝆} + \frac{1}{ρ} \frac{\partial A_{z}}{\partial φ} \hat{𝐳} \otimes \hat{𝝋} + \frac{\partial A_{z}}{\partial z} \hat{𝐳} \otimes \hat{𝐳} \end{matrix}$	$\begin{matrix} \frac{\partial A_{r}}{\partial r} \hat{𝐫} \otimes \hat{𝐫} + (\frac{1}{r} \frac{\partial A_{r}}{\partial θ} - \frac{A_{θ}}{r}) \hat{𝐫} \otimes \hat{𝜽} + (\frac{1}{r \sin θ} \frac{\partial A_{r}}{\partial φ} - \frac{A_{φ}}{r}) \hat{𝐫} \otimes \hat{𝝋} \\ + & \frac{\partial A_{θ}}{\partial r} \hat{𝜽} \otimes \hat{𝐫} + (\frac{1}{r} \frac{\partial A_{θ}}{\partial θ} + \frac{A_{r}}{r}) \hat{𝜽} \otimes \hat{𝜽} + (\frac{1}{r \sin θ} \frac{\partial A_{θ}}{\partial φ} - \cot θ \frac{A_{φ}}{r}) \hat{𝜽} \otimes \hat{𝝋} \\ + & \frac{\partial A_{φ}}{\partial r} \hat{𝝋} \otimes \hat{𝐫} + \frac{1}{r} \frac{\partial A_{φ}}{\partial θ} \hat{𝝋} \otimes \hat{𝜽} + (\frac{1}{r \sin θ} \frac{\partial A_{φ}}{\partial φ} + \cot θ \frac{A_{θ}}{r} + \frac{A_{r}}{r}) \hat{𝝋} \otimes \hat{𝝋} \end{matrix}$
Vector Laplacian $\nabla 2 A \equiv ∆ A$ ^[2]	$\nabla^{2} A_{x} \hat{𝐱} + \nabla^{2} A_{y} \hat{𝐲} + \nabla^{2} A_{z} \hat{𝐳}$	$\begin{matrix} (\nabla^{2} A_{ρ} - \frac{A_{ρ}}{ρ^{2}} - \frac{2}{ρ^{2}} \frac{\partial A_{φ}}{\partial φ}) & \hat{𝝆} \\ + (\nabla^{2} A_{φ} - \frac{A_{φ}}{ρ^{2}} + \frac{2}{ρ^{2}} \frac{\partial A_{ρ}}{\partial φ}) & \hat{𝝋} \\ + \nabla^{2} A_{z} & \hat{𝐳} \end{matrix}$	$\begin{matrix} (\nabla^{2} A_{r} - \frac{2 A_{r}}{r^{2}} - \frac{2}{r^{2} \sin θ} \frac{\partial (A_{θ} \sin θ)}{\partial θ} - \frac{2}{r^{2} \sin θ} \frac{\partial A_{φ}}{\partial φ}) & \hat{𝐫} \\ + (\nabla^{2} A_{θ} - \frac{A_{θ}}{r^{2} \sin^{2} θ} + \frac{2}{r^{2}} \frac{\partial A_{r}}{\partial θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} \frac{\partial A_{φ}}{\partial φ}) & \hat{𝜽} \\ + (\nabla^{2} A_{φ} - \frac{A_{φ}}{r^{2} \sin^{2} θ} + \frac{2}{r^{2} \sin θ} \frac{\partial A_{r}}{\partial φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} \frac{\partial A_{θ}}{\partial φ}) & \hat{𝝋} \end{matrix}$
Directional derivative $(A \cdot \nabla) B$ ^[3]	$𝐀 \cdot \nabla B_{x} \hat{𝐱} + 𝐀 \cdot \nabla B_{y} \hat{𝐲} + 𝐀 \cdot \nabla B_{z} \hat{𝐳}$	$\begin{matrix} (A_{ρ} \frac{\partial B_{ρ}}{\partial ρ} + \frac{A_{φ}}{ρ} \frac{\partial B_{ρ}}{\partial φ} + A_{z} \frac{\partial B_{ρ}}{\partial z} - \frac{A_{φ} B_{φ}}{ρ}) & \hat{𝝆} \\ + (A_{ρ} \frac{\partial B_{φ}}{\partial ρ} + \frac{A_{φ}}{ρ} \frac{\partial B_{φ}}{\partial φ} + A_{z} \frac{\partial B_{φ}}{\partial z} + \frac{A_{φ} B_{ρ}}{ρ}) & \hat{𝝋} \\ + (A_{ρ} \frac{\partial B_{z}}{\partial ρ} + \frac{A_{φ}}{ρ} \frac{\partial B_{z}}{\partial φ} + A_{z} \frac{\partial B_{z}}{\partial z}) & \hat{𝐳} \end{matrix}$	$\begin{matrix} (A_{r} \frac{\partial B_{r}}{\partial r} + \frac{A_{θ}}{r} \frac{\partial B_{r}}{\partial θ} + \frac{A_{φ}}{r \sin θ} \frac{\partial B_{r}}{\partial φ} - \frac{A_{θ} B_{θ} + A_{φ} B_{φ}}{r}) & \hat{𝐫} \\ + (A_{r} \frac{\partial B_{θ}}{\partial r} + \frac{A_{θ}}{r} \frac{\partial B_{θ}}{\partial θ} + \frac{A_{φ}}{r \sin θ} \frac{\partial B_{θ}}{\partial φ} + \frac{A_{θ} B_{r}}{r} - \frac{A_{φ} B_{φ} \cot θ}{r}) & \hat{𝜽} \\ + (A_{r} \frac{\partial B_{φ}}{\partial r} + \frac{A_{θ}}{r} \frac{\partial B_{φ}}{\partial θ} + \frac{A_{φ}}{r \sin θ} \frac{\partial B_{φ}}{\partial φ} + \frac{A_{φ} B_{r}}{r} + \frac{A_{φ} B_{θ} \cot θ}{r}) & \hat{𝝋} \end{matrix}$
Tensor divergence $\nabla \cdot T$ ^γ	$\begin{matrix} (\frac{\partial T_{x x}}{\partial x} + \frac{\partial T_{y x}}{\partial y} + \frac{\partial T_{z x}}{\partial z}) & \hat{𝐱} \\ + (\frac{\partial T_{x y}}{\partial x} + \frac{\partial T_{y y}}{\partial y} + \frac{\partial T_{z y}}{\partial z}) & \hat{𝐲} \\ + (\frac{\partial T_{x z}}{\partial x} + \frac{\partial T_{y z}}{\partial y} + \frac{\partial T_{z z}}{\partial z}) & \hat{𝐳} \end{matrix}$	$\begin{matrix} [\frac{\partial T_{ρ ρ}}{\partial ρ} + \frac{1}{ρ} \frac{\partial T_{φ ρ}}{\partial φ} + \frac{\partial T_{z ρ}}{\partial z} + \frac{1}{ρ} (T_{ρ ρ} - T_{φ φ})] & \hat{𝝆} \\ + [\frac{\partial T_{ρ φ}}{\partial ρ} + \frac{1}{ρ} \frac{\partial T_{φ φ}}{\partial φ} + \frac{\partial T_{z φ}}{\partial z} + \frac{1}{ρ} (T_{ρ φ} + T_{φ ρ})] & \hat{𝝋} \\ + [\frac{\partial T_{ρ z}}{\partial ρ} + \frac{1}{ρ} \frac{\partial T_{φ z}}{\partial φ} + \frac{\partial T_{z z}}{\partial z} + \frac{T_{ρ z}}{ρ}] & \hat{𝐳} \end{matrix}$	$\begin{matrix} [\frac{\partial T_{r r}}{\partial r} + 2 \frac{T_{r r}}{r} + \frac{1}{r} \frac{\partial T_{θ r}}{\partial θ} + \frac{\cot θ}{r} T_{θ r} + \frac{1}{r \sin θ} \frac{\partial T_{φ r}}{\partial φ} - \frac{1}{r} (T_{θ θ} + T_{φ φ})] & \hat{𝐫} \\ + [\frac{\partial T_{r θ}}{\partial r} + 2 \frac{T_{r θ}}{r} + \frac{1}{r} \frac{\partial T_{θ θ}}{\partial θ} + \frac{\cot θ}{r} T_{θ θ} + \frac{1}{r \sin θ} \frac{\partial T_{φ θ}}{\partial φ} + \frac{T_{θ r}}{r} - \frac{\cot θ}{r} T_{φ φ}] & \hat{𝜽} \\ + [\frac{\partial T_{r φ}}{\partial r} + 2 \frac{T_{r φ}}{r} + \frac{1}{r} \frac{\partial T_{θ φ}}{\partial θ} + \frac{1}{r \sin θ} \frac{\partial T_{φ φ}}{\partial φ} + \frac{T_{φ r}}{r} + \frac{\cot θ}{r} (T_{θ φ} + T_{φ θ})] & \hat{𝝋} \end{matrix}$

^α This page uses $θ$ for the polar angle and $φ$ for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses $θ$ for the azimuthal angle and $φ$ for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch $θ$ and $φ$ in the formulae shown in the table above.
^β Defined in Cartesian coordinates as $\partial_{i} 𝐀 \otimes 𝐞_{i}$ . An alternative definition is $𝐞_{i} \otimes \partial_{i} 𝐀$ .
^γ Defined in Cartesian coordinates as $𝐞_{i} \cdot \partial_{i} 𝐓$ . An alternative definition is $\partial_{i} 𝐓 \cdot 𝐞_{i}$ .

Differential elements

Operation	Cartesian coordinates $(x, y, z)$	Cylindrical coordinates $(ρ, φ, z)$	Spherical coordinates^α $(r, θ, φ)$
Differential displacement $d ℓ$ ^[1]	$d x \hat{𝐱} + d y \hat{𝐲} + d z \hat{𝐳}$	$d ρ \hat{𝝆} + ρ d φ \hat{𝝋} + d z \hat{𝐳}$	$d r \hat{𝐫} + r d θ \hat{𝜽} + r \sin θ d φ \hat{𝝋}$
Differential normal area $d S$	$\begin{matrix} d y d z & \hat{𝐱} \\ + d x d z & \hat{𝐲} \\ + d x d y & \hat{𝐳} \end{matrix}$	$\begin{matrix} ρ d φ d z & \hat{𝝆} \\ + d ρ d z & \hat{𝝋} \\ + ρ d ρ d φ & \hat{𝐳} \end{matrix}$	$\begin{matrix} r^{2} \sin θ d θ d φ & \hat{𝐫} \\ + r \sin θ d r d φ & \hat{𝜽} \\ + r d r d θ & \hat{𝝋} \end{matrix}$
Differential volume $dV$ ^[1]	$d x d y d z$	$ρ d ρ d φ d z$	$r^{2} \sin θ d r d θ d φ$

Calculation rules

$div grad f \equiv \nabla \cdot \nabla f \equiv \nabla^{2} f$
$curl grad f \equiv \nabla \times \nabla f = 𝟎$
$div curl 𝐀 \equiv \nabla \cdot (\nabla \times 𝐀) = 0$
$curl curl 𝐀 \equiv \nabla \times (\nabla \times 𝐀) = \nabla (\nabla \cdot 𝐀) - \nabla^{2} 𝐀$ (Lagrange's formula for del)
$\nabla^{2} (f g) = f \nabla^{2} g + 2 \nabla f \cdot \nabla g + g \nabla^{2} f$
$\nabla^{2} (𝐏 \cdot 𝐐) = 𝐐 \cdot \nabla^{2} 𝐏 - 𝐏 \cdot \nabla^{2} 𝐐 + 2 \nabla \cdot [(𝐏 \cdot \nabla) 𝐐 + 𝐏 \times \nabla \times 𝐐]$ (From ^[4] )

Cartesian derivation

File:Nabla cartesian.svg

$\begin{matrix} div 𝐀 & = \lim_{V \to 0} \frac{\iint_{\partial V} 𝐀 \cdot d 𝐒}{\underset{V}{∭} d V} \\ = \frac{[A_{x} (x + d x) - A_{x} (x)] d y d z + [A_{y} (y + d y) - A_{y} (y)] d x d z + [A_{z} (z + d z) - A_{z} (z)] d x d y}{d x d y d z} \\ = \frac{\partial A_{x}}{\partial x} + \frac{\partial A_{y}}{\partial y} + \frac{\partial A_{z}}{\partial z} \end{matrix}$

$\begin{matrix} (curl 𝐀)_{x} & = \lim_{S^{⊥ \hat{𝐱}} \to 0} \frac{\int_{\partial S} 𝐀 \cdot d ℓ}{\iint_{S} d S} \\ = \frac{[A_{z} (y + d y) - A_{z} (y)] d z - [A_{y} (z + d z) - A_{y} (z)] d y}{d y d z} \\ = \frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z} \end{matrix}$

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

Cylindrical derivation

File:Nabla cylindrical2.svg

$\begin{matrix} div 𝐀 & = \lim_{V \to 0} \frac{\iint_{\partial V} 𝐀 \cdot d 𝐒}{\underset{V}{∭} d V} \\ = \frac{[A_{ρ} (ρ + d ρ) (ρ + d ρ) - A_{ρ} (ρ) ρ] d ϕ d z + [A_{ϕ} (ϕ + d ϕ) - A_{ϕ} (ϕ)] d ρ d z + [A_{z} (z + d z) - A_{z} (z)] d ρ (ρ + d ρ / 2) d ϕ}{ρ d ϕ d ρ d z} \\ = \frac{1}{ρ} \frac{\partial (ρ A_{ρ})}{\partial ρ} + \frac{1}{ρ} \frac{\partial A_{ϕ}}{\partial ϕ} + \frac{\partial A_{z}}{\partial z} \end{matrix}$

$\begin{matrix} (curl 𝐀)_{ρ} & = \lim_{S^{⊥ \hat{𝝆}} \to 0} \frac{\int_{\partial S} 𝐀 \cdot d ℓ}{\iint_{S} d S} \\ = \frac{A_{ϕ} (z) (ρ + d ρ) d ϕ - A_{ϕ} (z + d z) (ρ + d ρ) d ϕ + A_{z} (ϕ + d ϕ) d z - A_{z} (ϕ) d z}{(ρ + d ρ) d ϕ d z} \\ = - \frac{\partial A_{ϕ}}{\partial z} + \frac{1}{ρ} \frac{\partial A_{z}}{\partial ϕ} \end{matrix}$

$\begin{matrix} (curl 𝐀)_{ϕ} & = \lim_{S^{⊥ \hat{𝝓}} \to 0} \frac{\int_{\partial S} 𝐀 \cdot d ℓ}{\iint_{S} d S} \\ = \frac{A_{z} (ρ) d z - A_{z} (ρ + d ρ) d z + A_{ρ} (z + d z) d ρ - A_{ρ} (z) d ρ}{d ρ d z} \\ = - \frac{\partial A_{z}}{\partial ρ} + \frac{\partial A_{ρ}}{\partial z} \end{matrix}$

$\begin{matrix} (curl 𝐀)_{z} & = \lim_{S^{⊥ \hat{𝒛}} \to 0} \frac{\int_{\partial S} 𝐀 \cdot d ℓ}{\iint_{S} d S} \\ = \frac{A_{ρ} (ϕ) d ρ - A_{ρ} (ϕ + d ϕ) d ρ + A_{ϕ} (ρ + d ρ) (ρ + d ρ) d ϕ - A_{ϕ} (ρ) ρ d ϕ}{ρ d ρ d ϕ} \\ = - \frac{1}{ρ} \frac{\partial A_{ρ}}{\partial ϕ} + \frac{1}{ρ} \frac{\partial (ρ A_{ϕ})}{\partial ρ} \end{matrix}$

$\begin{matrix} curl 𝐀 & = (curl 𝐀)_{ρ} \hat{𝝆} + (curl 𝐀)_{ϕ} \hat{𝝓} + (curl 𝐀)_{z} \hat{𝒛} \\ = (\frac{1}{ρ} \frac{\partial A_{z}}{\partial ϕ} - \frac{\partial A_{ϕ}}{\partial z}) \hat{𝝆} + (\frac{\partial A_{ρ}}{\partial z} - \frac{\partial A_{z}}{\partial ρ}) \hat{𝝓} + \frac{1}{ρ} (\frac{\partial (ρ A_{ϕ})}{\partial ρ} - \frac{\partial A_{ρ}}{\partial ϕ}) \hat{𝒛} \end{matrix}$

Spherical derivation

File:Nabla spherical2.svg $\begin{matrix} div 𝐀 & = \lim_{V \to 0} \frac{\iint_{\partial V} 𝐀 \cdot d 𝐒}{\underset{V}{∭} d V} \\ = \frac{[A_{r} (r + d r) (r + d r)^{2} - A_{r} (r) r^{2}] \sin θ d θ d ϕ + [A_{θ} (θ + d θ) \sin (θ + d θ) - A_{θ} (θ) \sin θ] r d r d ϕ + [A_{ϕ} (ϕ + d ϕ) - A_{ϕ} (ϕ)] r d r d θ}{d r r d θ r \sin θ d ϕ} \\ = \frac{1}{r^{2}} \frac{\partial (r^{2} A_{r})}{\partial r} + \frac{1}{r \sin θ} \frac{\partial (A_{θ} \sin θ)}{\partial θ} + \frac{1}{r \sin θ} \frac{\partial A_{ϕ}}{\partial ϕ} \end{matrix}$

$\begin{matrix} (curl 𝐀)_{r} & = \lim_{S^{⊥ \hat{𝒓}} \to 0} \frac{\int_{\partial S} 𝐀 \cdot d ℓ}{\iint_{S} d S} \\ = \frac{A_{θ} (ϕ) r d θ + A_{ϕ} (θ + d θ) r \sin (θ + d θ) d ϕ - A_{θ} (ϕ + d ϕ) r d θ - A_{ϕ} (θ) r \sin (θ) d ϕ}{r d θ r \sin θ d ϕ} \\ = \frac{1}{r \sin θ} \frac{\partial (A_{ϕ} \sin θ)}{\partial θ} - \frac{1}{r \sin θ} \frac{\partial A_{θ}}{\partial ϕ} \end{matrix}$

$\begin{matrix} (curl 𝐀)_{θ} & = \lim_{S^{⊥ \hat{𝜽}} \to 0} \frac{\int_{\partial S} 𝐀 \cdot d ℓ}{\iint_{S} d S} \\ = \frac{A_{ϕ} (r) r \sin θ d ϕ + A_{r} (ϕ + d ϕ) d r - A_{ϕ} (r + d r) (r + d r) \sin θ d ϕ - A_{r} (ϕ) d r}{d r r \sin θ d ϕ} \\ = \frac{1}{r \sin θ} \frac{\partial A_{r}}{\partial ϕ} - \frac{1}{r} \frac{\partial (r A_{ϕ})}{\partial r} \end{matrix}$

$\begin{matrix} (curl 𝐀)_{ϕ} & = \lim_{S^{⊥ \hat{𝝓}} \to 0} \frac{\int_{\partial S} 𝐀 \cdot d ℓ}{\iint_{S} d S} \\ = \frac{A_{r} (θ) d r + A_{θ} (r + d r) (r + d r) d θ - A_{r} (θ + d θ) d r - A_{θ} (r) r d θ}{r d r d θ} \\ = \frac{1}{r} \frac{\partial (r A_{θ})}{\partial r} - \frac{1}{r} \frac{\partial A_{r}}{\partial θ} \end{matrix}$

$\begin{matrix} curl 𝐀 & = (curl 𝐀)_{r} \hat{𝒓} + (curl 𝐀)_{θ} \hat{𝜽} + (curl 𝐀)_{ϕ} \hat{𝝓} \\ = \frac{1}{r \sin θ} (\frac{\partial (A_{ϕ} \sin θ)}{\partial θ} - \frac{\partial A_{θ}}{\partial ϕ}) \hat{𝒓} + \frac{1}{r} (\frac{1}{\sin θ} \frac{\partial A_{r}}{\partial ϕ} - \frac{\partial (r A_{ϕ})}{\partial r}) \hat{𝜽} + \frac{1}{r} (\frac{\partial (r A_{θ})}{\partial r} - \frac{\partial A_{r}}{\partial θ}) \hat{𝝓} \end{matrix}$

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

For two sets of coordinate systems $u_{i}$ and $v_{j}$ , according to chain rule, $\begin{matrix} d 𝐫 & = \sum_{i} \frac{\partial 𝐫}{\partial u_{i}} d u_{i} = \sum_{i} \frac{\partial s}{\partial u_{i}} {\hat{𝐮}}_{i} d u_{i} = \sum_{j} \frac{\partial s}{\partial v_{j}} {\hat{𝐯}}_{j} d v_{j} \\ = \sum_{j} \frac{\partial s}{\partial v_{j}} {\hat{𝐯}}_{j} \sum_{i} \frac{\partial v_{j}}{\partial u_{i}} d u_{i} \\ = \sum_{i} \sum_{j} \frac{\partial s}{\partial v_{j}} \frac{\partial v_{j}}{\partial u_{i}} {\hat{𝐯}}_{j} d u_{i} . \end{matrix}$

Now, we isolate the $i$ ^th component. For $i \neq k$ , let $d u_{k} = 0$ . Then divide on both sides by $d u_{i}$ to get: $\frac{\partial s}{\partial u_{i}} {\hat{𝐮}}_{i} = \sum_{j} \frac{\partial s}{\partial v_{j}} \frac{\partial v_{j}}{\partial u_{i}} {\hat{𝐯}}_{j} .$

References

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^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

External links

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

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[2] Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).

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Del in cylindrical and spherical coordinates

Contents

Notes

Coordinate conversions

Unit vector conversions

Del formula

Differential elements

Calculation rules

Cartesian derivation

Cylindrical derivation

Spherical derivation

Unit vector conversion formula

See also

References

External links

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Del in cylindrical and spherical coordinates

Notes

Coordinate conversions

Unit vector conversions

Del formula

Differential elements

Calculation rules

Cartesian derivation

Cylindrical derivation

Spherical derivation

Unit vector conversion formula

See also

References

External links

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