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

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	$\begin{aligned} \hat{𝐱} & = \hat{𝐱} \\ \hat{𝐲} & = \hat{𝐲} \\ \hat{𝐳} & = \hat{𝐳} \end{aligned}$	$\begin{aligned} \hat{𝐱} & = \cos φ \hat{ρ} - \sin φ \hat{φ} \\ \hat{𝐲} & = \sin φ \hat{ρ} + \cos φ \hat{φ} \\ \hat{𝐳} & = \hat{𝐳} \end{aligned}$	$\begin{aligned} \hat{𝐱} & = \sin θ \cos φ \hat{𝐫} + \cos θ \cos φ \hat{θ} - \sin φ \hat{φ} \\ \hat{𝐲} & = \sin θ \sin φ \hat{𝐫} + \cos θ \sin φ \hat{θ} + \cos φ \hat{φ} \\ \hat{𝐳} & = \cos θ \hat{𝐫} - \sin θ \hat{θ} \end{aligned}$
Cylindrical	$\begin{aligned} \hat{ρ} & = \frac{x \hat{𝐱} + y \hat{𝐲}}{\sqrt{x^{2} + y^{2}}} \\ \hat{φ} & = \frac{- y \hat{𝐱} + x \hat{𝐲}}{\sqrt{x^{2} + y^{2}}} \\ \hat{𝐳} & = \hat{𝐳} \end{aligned}$	$\begin{aligned} \hat{ρ} & = \hat{ρ} \\ \hat{φ} & = \hat{φ} \\ \hat{𝐳} & = \hat{𝐳} \end{aligned}$	$\begin{aligned} \hat{ρ} & = \sin θ \hat{𝐫} + \cos θ \hat{θ} \\ \hat{φ} & = \hat{φ} \\ \hat{𝐳} & = \cos θ \hat{𝐫} - \sin θ \hat{θ} \end{aligned}$
Spherical	$\begin{aligned} \hat{𝐫} & = \frac{x \hat{𝐱} + y \hat{𝐲} + z \hat{𝐳}}{\sqrt{x^{2} + y^{2} + z^{2}}} \\ \hat{θ} & = \frac{(x \hat{𝐱} + y \hat{𝐲}) z - (x^{2} + y^{2}) \hat{𝐳}}{\sqrt{x^{2} + y^{2} + z^{2}} \sqrt{x^{2} + y^{2}}} \\ \hat{φ} & = \frac{- y \hat{𝐱} + x \hat{𝐲}}{\sqrt{x^{2} + y^{2}}} \end{aligned}$	$\begin{aligned} \hat{𝐫} & = \frac{ρ \hat{ρ} + z \hat{𝐳}}{\sqrt{ρ^{2} + z^{2}}} \\ \hat{θ} & = \frac{z \hat{ρ} - ρ \hat{𝐳}}{\sqrt{ρ^{2} + z^{2}}} \\ \hat{φ} & = \hat{φ} \end{aligned}$	$\begin{aligned} \hat{𝐫} & = \hat{𝐫} \\ \hat{θ} & = \hat{θ} \\ \hat{φ} & = \hat{φ} \end{aligned}$

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

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]	$\begin{aligned} (\frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z}) & \hat{𝐱} \\ + (\frac{\partial A_{x}}{\partial z} - \frac{\partial A_{z}}{\partial x}) & \hat{𝐲} \\ + (\frac{\partial A_{y}}{\partial x} - \frac{\partial A_{x}}{\partial y}) & \hat{𝐳} \end{aligned}$	$\begin{aligned} (\frac{1}{ρ} \frac{\partial A_{z}}{\partial φ} - \frac{\partial A_{φ}}{\partial z}) & \hat{ρ} \\ + (\frac{\partial A_{ρ}}{\partial z} - \frac{\partial A_{z}}{\partial ρ}) & \hat{φ} \\ + \frac{1}{ρ} (\frac{\partial (ρ A_{φ})}{\partial ρ} - \frac{\partial A_{ρ}}{\partial φ}) & \hat{𝐳} \end{aligned}$	$\begin{aligned} \frac{1}{r \sin θ} (\frac{\partial}{\partial θ} (A_{φ} \sin θ) - \frac{\partial A_{θ}}{\partial φ}) & \hat{𝐫} \\ + \frac{1}{r} (\frac{1}{\sin θ} \frac{\partial A_{r}}{\partial φ} - \frac{\partial}{\partial r} (r A_{φ})) & \hat{θ} \\ + \frac{1}{r} (\frac{\partial}{\partial r} (r A_{θ}) - \frac{\partial A_{r}}{\partial θ}) & \hat{φ} \end{aligned}$
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	$\begin{aligned} \frac{\partial A_{x}}{\partial x} \hat{𝐱} \otimes \hat{𝐱} + \frac{\partial A_{x}}{\partial y} \hat{𝐱} \otimes \hat{𝐲} + \frac{\partial A_{x}}{\partial z} \hat{𝐱} \otimes \hat{𝐳} \\ + & \frac{\partial A_{y}}{\partial x} \hat{𝐲} \otimes \hat{𝐱} + \frac{\partial A_{y}}{\partial y} \hat{𝐲} \otimes \hat{𝐲} + \frac{\partial A_{y}}{\partial z} \hat{𝐲} \otimes \hat{𝐳} \\ + & \frac{\partial A_{z}}{\partial x} \hat{𝐳} \otimes \hat{𝐱} + \frac{\partial A_{z}}{\partial y} \hat{𝐳} \otimes \hat{𝐲} + \frac{\partial A_{z}}{\partial z} \hat{𝐳} \otimes \hat{𝐳} \end{aligned}$	$\begin{aligned} \frac{\partial A_{ρ}}{\partial ρ} \hat{ρ} \otimes \hat{ρ} + (\frac{1}{ρ} \frac{\partial A_{ρ}}{\partial φ} - \frac{A_{φ}}{ρ}) \hat{ρ} \otimes \hat{φ} + \frac{\partial A_{ρ}}{\partial z} \hat{ρ} \otimes \hat{𝐳} \\ + & \frac{\partial A_{φ}}{\partial ρ} \hat{φ} \otimes \hat{ρ} + (\frac{1}{ρ} \frac{\partial A_{φ}}{\partial φ} + \frac{A_{ρ}}{ρ}) \hat{φ} \otimes \hat{φ} + \frac{\partial A_{φ}}{\partial z} \hat{φ} \otimes \hat{𝐳} \\ + & \frac{\partial A_{z}}{\partial ρ} \hat{𝐳} \otimes \hat{ρ} + \frac{1}{ρ} \frac{\partial A_{z}}{\partial φ} \hat{𝐳} \otimes \hat{φ} + \frac{\partial A_{z}}{\partial z} \hat{𝐳} \otimes \hat{𝐳} \end{aligned}$	$\begin{aligned} \frac{\partial A_{r}}{\partial r} \hat{𝐫} \otimes \hat{𝐫} + (\frac{1}{r} \frac{\partial A_{r}}{\partial θ} - \frac{A_{θ}}{r}) \hat{𝐫} \otimes \hat{θ} + (\frac{1}{r \sin θ} \frac{\partial A_{r}}{\partial φ} - \frac{A_{φ}}{r}) \hat{𝐫} \otimes \hat{φ} \\ + & \frac{\partial A_{θ}}{\partial r} \hat{θ} \otimes \hat{𝐫} + (\frac{1}{r} \frac{\partial A_{θ}}{\partial θ} + \frac{A_{r}}{r}) \hat{θ} \otimes \hat{θ} + (\frac{1}{r \sin θ} \frac{\partial A_{θ}}{\partial φ} - \cot θ \frac{A_{φ}}{r}) \hat{θ} \otimes \hat{φ} \\ + & \frac{\partial A_{φ}}{\partial r} \hat{φ} \otimes \hat{𝐫} + \frac{1}{r} \frac{\partial A_{φ}}{\partial θ} \hat{φ} \otimes \hat{θ} + (\frac{1}{r \sin θ} \frac{\partial A_{φ}}{\partial φ} + \cot θ \frac{A_{θ}}{r} + \frac{A_{r}}{r}) \hat{φ} \otimes \hat{φ} \end{aligned}$
Vector Laplacian $\nabla 2 A \equiv ∆ A$ ^[2]	$\nabla^{2} A_{x} \hat{𝐱} + \nabla^{2} A_{y} \hat{𝐲} + \nabla^{2} A_{z} \hat{𝐳}$	$\begin{aligned} (\nabla^{2} A_{ρ} - \frac{A_{ρ}}{ρ^{2}} - \frac{2}{ρ^{2}} \frac{\partial A_{φ}}{\partial φ}) & \hat{ρ} \\ + (\nabla^{2} A_{φ} - \frac{A_{φ}}{ρ^{2}} + \frac{2}{ρ^{2}} \frac{\partial A_{ρ}}{\partial φ}) & \hat{φ} \\ + \nabla^{2} A_{z} & \hat{𝐳} \end{aligned}$	$\begin{aligned} (\nabla^{2} A_{r} - \frac{2 A_{r}}{r^{2}} - \frac{2}{r^{2} \sin θ} \frac{\partial (A_{θ} \sin θ)}{\partial θ} - \frac{2}{r^{2} \sin θ} \frac{\partial A_{φ}}{\partial φ}) & \hat{𝐫} \\ + (\nabla^{2} A_{θ} - \frac{A_{θ}}{r^{2} \sin^{2} θ} + \frac{2}{r^{2}} \frac{\partial A_{r}}{\partial θ} - \frac{2 \cos θ}{r^{2} \sin^{2} θ} \frac{\partial A_{φ}}{\partial φ}) & \hat{θ} \\ + (\nabla^{2} A_{φ} - \frac{A_{φ}}{r^{2} \sin^{2} θ} + \frac{2}{r^{2} \sin θ} \frac{\partial A_{r}}{\partial φ} + \frac{2 \cos θ}{r^{2} \sin^{2} θ} \frac{\partial A_{θ}}{\partial φ}) & \hat{φ} \end{aligned}$
Directional derivative $(A \cdot \nabla) B$ ^[3]	$𝐀 \cdot \nabla B_{x} \hat{𝐱} + 𝐀 \cdot \nabla B_{y} \hat{𝐲} + 𝐀 \cdot \nabla B_{z} \hat{𝐳}$	$\begin{aligned} (A_{ρ} \frac{\partial B_{ρ}}{\partial ρ} + \frac{A_{φ}}{ρ} \frac{\partial B_{ρ}}{\partial φ} + A_{z} \frac{\partial B_{ρ}}{\partial z} - \frac{A_{φ} B_{φ}}{ρ}) & \hat{ρ} \\ + (A_{ρ} \frac{\partial B_{φ}}{\partial ρ} + \frac{A_{φ}}{ρ} \frac{\partial B_{φ}}{\partial φ} + A_{z} \frac{\partial B_{φ}}{\partial z} + \frac{A_{φ} B_{ρ}}{ρ}) & \hat{φ} \\ + (A_{ρ} \frac{\partial B_{z}}{\partial ρ} + \frac{A_{φ}}{ρ} \frac{\partial B_{z}}{\partial φ} + A_{z} \frac{\partial B_{z}}{\partial z}) & \hat{𝐳} \end{aligned}$	$\begin{aligned} (A_{r} \frac{\partial B_{r}}{\partial r} + \frac{A_{θ}}{r} \frac{\partial B_{r}}{\partial θ} + \frac{A_{φ}}{r \sin θ} \frac{\partial B_{r}}{\partial φ} - \frac{A_{θ} B_{θ} + A_{φ} B_{φ}}{r}) & \hat{𝐫} \\ + (A_{r} \frac{\partial B_{θ}}{\partial r} + \frac{A_{θ}}{r} \frac{\partial B_{θ}}{\partial θ} + \frac{A_{φ}}{r \sin θ} \frac{\partial B_{θ}}{\partial φ} + \frac{A_{θ} B_{r}}{r} - \frac{A_{φ} B_{φ} \cot θ}{r}) & \hat{θ} \\ + (A_{r} \frac{\partial B_{φ}}{\partial r} + \frac{A_{θ}}{r} \frac{\partial B_{φ}}{\partial θ} + \frac{A_{φ}}{r \sin θ} \frac{\partial B_{φ}}{\partial φ} + \frac{A_{φ} B_{r}}{r} + \frac{A_{φ} B_{θ} \cot θ}{r}) & \hat{φ} \end{aligned}$
Tensor divergence $\nabla \cdot T$ ^γ	$\begin{aligned} (\frac{\partial T_{x x}}{\partial x} + \frac{\partial T_{y x}}{\partial y} + \frac{\partial T_{z x}}{\partial z}) & \hat{𝐱} \\ + (\frac{\partial T_{x y}}{\partial x} + \frac{\partial T_{y y}}{\partial y} + \frac{\partial T_{z y}}{\partial z}) & \hat{𝐲} \\ + (\frac{\partial T_{x z}}{\partial x} + \frac{\partial T_{y z}}{\partial y} + \frac{\partial T_{z z}}{\partial z}) & \hat{𝐳} \end{aligned}$	$\begin{aligned} [\frac{\partial T_{ρ ρ}}{\partial ρ} + \frac{1}{ρ} \frac{\partial T_{φ ρ}}{\partial φ} + \frac{\partial T_{z ρ}}{\partial z} + \frac{1}{ρ} (T_{ρ ρ} - T_{φ φ})] & \hat{ρ} \\ + [\frac{\partial T_{ρ φ}}{\partial ρ} + \frac{1}{ρ} \frac{\partial T_{φ φ}}{\partial φ} + \frac{\partial T_{z φ}}{\partial z} + \frac{1}{ρ} (T_{ρ φ} + T_{φ ρ})] & \hat{φ} \\ + [\frac{\partial T_{ρ z}}{\partial ρ} + \frac{1}{ρ} \frac{\partial T_{φ z}}{\partial φ} + \frac{\partial T_{z z}}{\partial z} + \frac{T_{ρ z}}{ρ}] & \hat{𝐳} \end{aligned}$	$\begin{aligned} [\frac{\partial T_{r r}}{\partial r} + 2 \frac{T_{r r}}{r} + \frac{1}{r} \frac{\partial T_{θ r}}{\partial θ} + \frac{\cot θ}{r} T_{θ r} + \frac{1}{r \sin θ} \frac{\partial T_{φ r}}{\partial φ} - \frac{1}{r} (T_{θ θ} + T_{φ φ})] & \hat{𝐫} \\ + [\frac{\partial T_{r θ}}{\partial r} + 2 \frac{T_{r θ}}{r} + \frac{1}{r} \frac{\partial T_{θ θ}}{\partial θ} + \frac{\cot θ}{r} T_{θ θ} + \frac{1}{r \sin θ} \frac{\partial T_{φ θ}}{\partial φ} + \frac{T_{θ r}}{r} - \frac{\cot θ}{r} T_{φ φ}] & \hat{θ} \\ + [\frac{\partial T_{r φ}}{\partial r} + 2 \frac{T_{r φ}}{r} + \frac{1}{r} \frac{\partial T_{θ φ}}{\partial θ} + \frac{1}{r \sin θ} \frac{\partial T_{φ φ}}{\partial φ} + \frac{T_{φ r}}{r} + \frac{\cot θ}{r} (T_{θ φ} + T_{φ θ})] & \hat{φ} \end{aligned}$

^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 $d S$	$\begin{aligned} d y d z & \hat{𝐱} \\ + d x d z & \hat{𝐲} \\ + d x d y & \hat{𝐳} \end{aligned}$	$\begin{aligned} ρ d φ d z & \hat{ρ} \\ + d ρ d z & \hat{φ} \\ + ρ d ρ d φ & \hat{𝐳} \end{aligned}$	$\begin{aligned} r^{2} \sin θ d θ d φ & \hat{𝐫} \\ + r \sin θ d r d φ & \hat{θ} \\ + r d r d θ & \hat{φ} \end{aligned}$
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

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

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

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

Cylindrical derivation

File:Nabla cylindrical2.svg

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

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

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

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

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

Spherical derivation

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

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

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

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

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

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, $\begin{aligned} d 𝐫 & = \sum_{i} \frac{\partial 𝐫}{\partial u_{i}} d u_{i} = \sum_{i} \frac{\partial s}{\partial u_{i}} {\hat{𝐮}}_{i} d u_{i} = \sum_{j} \frac{\partial s}{\partial v_{j}} {\hat{𝐯}}_{j} d v_{j} \\ = \sum_{j} \frac{\partial s}{\partial v_{j}} {\hat{𝐯}}_{j} \sum_{i} \frac{\partial v_{j}}{\partial u_{i}} d u_{i} \\ = \sum_{i} \sum_{j} \frac{\partial s}{\partial v_{j}} \frac{\partial v_{j}}{\partial u_{i}} {\hat{𝐯}}_{j} d u_{i} . \end{aligned}$

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

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 ^1.7 Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2.
↑ Arfken, George; Weber, Hans; Harris, Frank (2012). Mathematical Methods for Physicists (Seventh ed.). Academic Press. p. 192. ISBN 9789381269558.
↑ Weisstein, Eric W.. "Convective Operator". Mathworld. http://mathworld.wolfram.com/ConvectiveOperator.html.
↑ Fernández-Guasti, M. (2012). "Green's Second Identity for Vector Fields". ISRN Mathematical Physics (Hindawi Limited) 2012: 1–7. doi:10.5402/2012/973968. ISSN 2090-4681.

External links

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

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Original source: https://en.wikipedia.org/wiki/Del in cylindrical and spherical coordinates. Read more

[griffiths-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 ^1.7 Griffiths, David J. (2012). Introduction to Electrodynamics. Pearson. ISBN 978-0-321-85656-2.

[2] Arfken, George; Weber, Hans; Harris, Frank (2012). Mathematical Methods for Physicists (Seventh ed.). Academic Press. p. 192. ISBN 9789381269558.

[Mathworld-3] Weisstein, Eric W.. "Convective Operator". Mathworld. http://mathworld.wolfram.com/ConvectiveOperator.html.

[4] Fernández-Guasti, M. (2012). "Green's Second Identity for Vector Fields". ISRN Mathematical Physics (Hindawi Limited) 2012: 1–7. doi:10.5402/2012/973968. ISSN 2090-4681.

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