Del in cylindrical and spherical coordinates

Template:Short description 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 Template:Nowrap can be used instead of the mathematical function Template:Nowrap owing to its domain and image. The classical arctan function has an image of Template:Nowrap, whereas atan2 is defined to have an image of Template:Nowrap.

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

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Table with the del operator in cartesian, cylindrical and spherical coordinates
Operation	Cartesian coordinates Template:Math	Cylindrical coordinates Template:Math	Spherical coordinates Template:Math, where Template:Math is the polar angle and Template:Math is the azimuthal angleTemplate:Ref
Vector field Template:Math	$A_{x} \hat{𝐱} + A_{y} \hat{𝐲} + A_{z} \hat{𝐳}$	$A_{ρ} \hat{𝝆} + A_{φ} \hat{𝝋} + A_{z} \hat{𝐳}$	$A_{r} \hat{𝐫} + A_{θ} \hat{𝜽} + A_{φ} \hat{𝝋}$
Gradient Template:Math^[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 Template:Math^[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 Template:Math^[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 Template:Math^[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 Template:MathTemplate:Ref	$\begin{matrix} \frac{\partial A_{x}}{\partial x} \hat{𝐱} \otimes \hat{𝐱} + \frac{\partial A_{x}}{\partial y} \hat{𝐱} \otimes \hat{𝐲} + \frac{\partial A_{x}}{\partial z} \hat{𝐱} \otimes \hat{𝐳} \\ + & \frac{\partial A_{y}}{\partial x} \hat{𝐲} \otimes \hat{𝐱} + \frac{\partial A_{y}}{\partial y} \hat{𝐲} \otimes \hat{𝐲} + \frac{\partial A_{y}}{\partial z} \hat{𝐲} \otimes \hat{𝐳} \\ + & \frac{\partial A_{z}}{\partial x} \hat{𝐳} \otimes \hat{𝐱} + \frac{\partial A_{z}}{\partial y} \hat{𝐳} \otimes \hat{𝐲} + \frac{\partial A_{z}}{\partial z} \hat{𝐳} \otimes \hat{𝐳} \end{matrix}$	$\begin{matrix} \frac{\partial A_{ρ}}{\partial ρ} \hat{𝝆} \otimes \hat{𝝆} + (\frac{1}{ρ} \frac{\partial A_{ρ}}{\partial φ} - \frac{A_{φ}}{ρ}) \hat{𝝆} \otimes \hat{𝝋} + \frac{\partial A_{ρ}}{\partial z} \hat{𝝆} \otimes \hat{𝐳} \\ + & \frac{\partial A_{φ}}{\partial ρ} \hat{𝝋} \otimes \hat{𝝆} + (\frac{1}{ρ} \frac{\partial A_{φ}}{\partial φ} + \frac{A_{ρ}}{ρ}) \hat{𝝋} \otimes \hat{𝝋} + \frac{\partial A_{φ}}{\partial z} \hat{𝝋} \otimes \hat{𝐳} \\ + & \frac{\partial A_{z}}{\partial ρ} \hat{𝐳} \otimes \hat{𝝆} + \frac{1}{ρ} \frac{\partial A_{z}}{\partial φ} \hat{𝐳} \otimes \hat{𝝋} + \frac{\partial A_{z}}{\partial z} \hat{𝐳} \otimes \hat{𝐳} \end{matrix}$	$\begin{matrix} \frac{\partial A_{r}}{\partial r} \hat{𝐫} \otimes \hat{𝐫} + (\frac{1}{r} \frac{\partial A_{r}}{\partial θ} - \frac{A_{θ}}{r}) \hat{𝐫} \otimes \hat{𝜽} + (\frac{1}{r \sin θ} \frac{\partial A_{r}}{\partial φ} - \frac{A_{φ}}{r}) \hat{𝐫} \otimes \hat{𝝋} \\ + & \frac{\partial A_{θ}}{\partial r} \hat{𝜽} \otimes \hat{𝐫} + (\frac{1}{r} \frac{\partial A_{θ}}{\partial θ} + \frac{A_{r}}{r}) \hat{𝜽} \otimes \hat{𝜽} + (\frac{1}{r \sin θ} \frac{\partial A_{θ}}{\partial φ} - \cot θ \frac{A_{φ}}{r}) \hat{𝜽} \otimes \hat{𝝋} \\ + & \frac{\partial A_{φ}}{\partial r} \hat{𝝋} \otimes \hat{𝐫} + \frac{1}{r} \frac{\partial A_{φ}}{\partial θ} \hat{𝝋} \otimes \hat{𝜽} + (\frac{1}{r \sin θ} \frac{\partial A_{φ}}{\partial φ} + \cot θ \frac{A_{θ}}{r} + \frac{A_{r}}{r}) \hat{𝝋} \otimes \hat{𝝋} \end{matrix}$
Vector Laplacian Template:Math^[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 Template:Math^[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 Template:MathTemplate:Ref	$\begin{matrix} (\frac{\partial T_{x x}}{\partial x} + \frac{\partial T_{y x}}{\partial y} + \frac{\partial T_{z x}}{\partial z}) & \hat{𝐱} \\ + (\frac{\partial T_{x y}}{\partial x} + \frac{\partial T_{y y}}{\partial y} + \frac{\partial T_{z y}}{\partial z}) & \hat{𝐲} \\ + (\frac{\partial T_{x z}}{\partial x} + \frac{\partial T_{y z}}{\partial y} + \frac{\partial T_{z z}}{\partial z}) & \hat{𝐳} \end{matrix}$	$\begin{matrix} [\frac{\partial T_{ρ ρ}}{\partial ρ} + \frac{1}{ρ} \frac{\partial T_{φ ρ}}{\partial φ} + \frac{\partial T_{z ρ}}{\partial z} + \frac{1}{ρ} (T_{ρ ρ} - T_{φ φ})] & \hat{𝝆} \\ + [\frac{\partial T_{ρ φ}}{\partial ρ} + \frac{1}{ρ} \frac{\partial T_{φ φ}}{\partial φ} + \frac{\partial T_{z φ}}{\partial z} + \frac{1}{ρ} (T_{ρ φ} + T_{φ ρ})] & \hat{𝝋} \\ + [\frac{\partial T_{ρ z}}{\partial ρ} + \frac{1}{ρ} \frac{\partial T_{φ z}}{\partial φ} + \frac{\partial T_{z z}}{\partial z} + \frac{T_{ρ z}}{ρ}] & \hat{𝐳} \end{matrix}$	$\begin{matrix} [\frac{\partial T_{r r}}{\partial r} + 2 \frac{T_{r r}}{r} + \frac{1}{r} \frac{\partial T_{θ r}}{\partial θ} + \frac{\cot θ}{r} T_{θ r} + \frac{1}{r \sin θ} \frac{\partial T_{φ r}}{\partial φ} - \frac{1}{r} (T_{θ θ} + T_{φ φ})] & \hat{𝐫} \\ + [\frac{\partial T_{r θ}}{\partial r} + 2 \frac{T_{r θ}}{r} + \frac{1}{r} \frac{\partial T_{θ θ}}{\partial θ} + \frac{\cot θ}{r} T_{θ θ} + \frac{1}{r \sin θ} \frac{\partial T_{φ θ}}{\partial φ} + \frac{T_{θ r}}{r} - \frac{\cot θ}{r} T_{φ φ}] & \hat{𝜽} \\ + [\frac{\partial T_{r φ}}{\partial r} + 2 \frac{T_{r φ}}{r} + \frac{1}{r} \frac{\partial T_{θ φ}}{\partial θ} + \frac{1}{r \sin θ} \frac{\partial T_{φ φ}}{\partial φ} + \frac{T_{φ r}}{r} + \frac{\cot θ}{r} (T_{θ φ} + T_{φ θ})] & \hat{𝝋} \end{matrix}$

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Differential elements

Operation	Cartesian coordinates Template:Math	Cylindrical coordinates Template:Math	Spherical coordinates Template:Ref Template:Math
Differential displacement Template:Math^[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 Template:Math	$\begin{matrix} d y d z & \hat{𝐱} \\ + d x d z & \hat{𝐲} \\ + d x d y & \hat{𝐳} \end{matrix}$	$\begin{matrix} ρ d φ d z & \hat{𝝆} \\ + d ρ d z & \hat{𝝋} \\ + ρ d ρ d φ & \hat{𝐳} \end{matrix}$	$\begin{matrix} r^{2} \sin θ d θ d φ & \hat{𝐫} \\ + r \sin θ d r d φ & \hat{𝜽} \\ + r d r d θ & \hat{𝝋} \end{matrix}$
Differential volume Template:Math^[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

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

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

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

Cylindrical derivation

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

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

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

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

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

Spherical derivation

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

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

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

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

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

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 Template:Mvar 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

Template:Reflist

External links

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

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 ^1.7 Template:Cite book
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[griffiths-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 ^1.7 Template:Cite book

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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

Navigation menu

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

Navigation menu

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