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.

## Contents

* 1 Notes
* 2 Coordinate conversions
* 3 Unit vector conversions
* 4 Del formula
* 4.1 Differential elements
* 4.2 Calculation rules
* 5 Cartesian derivation
* 6 Cylindrical derivation
* 7 Spherical derivation
* 8 Unit vector conversion formula
* 9 See also
* 10 References
* 11 External links

## 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∈[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∈[0,2π]: 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

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Cartesian  | Cylindrical  | Spherical
To  | Cartesian  | x=xy=yz=z | x=ρcos⁡φy=ρsin⁡φz=z | x=rsin⁡θcos⁡φy=rsin⁡θsin⁡φz=rcos⁡θ
Cylindrical  | ρ=x2+y2φ=arctan⁡(yx)z=z | ρ=ρφ=φz=z | ρ=rsin⁡θφ=φz=rcos⁡θ
Spherical  | r=x2+y2+z2θ=arctan⁡(x2+y2z)φ=arctan⁡(yx) | r=ρ2+z2θ=arctan⁡(ρz)φ=φ | r=rθ=θφ=φ

Note that the operation arctan⁡(AB) 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

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Cartesian  | 𝐱^=𝐱^𝐲^=𝐲^𝐳^=𝐳^ | 𝐱^=cos⁡φρ^−sin⁡φφ^𝐲^=sin⁡φρ^+cos⁡φφ^𝐳^=𝐳^ | 𝐱^=sin⁡θcos⁡φ𝐫^+cos⁡θcos⁡φθ^−sin⁡φφ^𝐲^=sin⁡θsin⁡φ𝐫^+cos⁡θsin⁡φθ^+cos⁡φφ^𝐳^=cos⁡θ𝐫^−sin⁡θθ^
Cylindrical  | ρ^=x𝐱^+y𝐲^x2+y2φ^=−y𝐱^+x𝐲^x2+y2𝐳^=𝐳^ | ρ^=ρ^φ^=φ^𝐳^=𝐳^ | ρ^=sin⁡θ𝐫^+cos⁡θθ^φ^=φ^𝐳^=cos⁡θ𝐫^−sin⁡θθ^
Spherical  | 𝐫^=x𝐱^+y𝐲^+z𝐳^x2+y2+z2θ^=(x𝐱^+y𝐲^)z−(x2+y2)𝐳^x2+y2+z2x2+y2φ^=−y𝐱^+x𝐲^x2+y2 | 𝐫^=ρρ^+z𝐳^ρ2+z2θ^=zρ^−ρ𝐳^ρ2+z2φ^=φ^ | 𝐫^=𝐫^θ^=θ^φ^=φ^
Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of source coordinates  | Cartesian  | Cylindrical  | Spherical

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Cartesian  | 𝐱^=𝐱^𝐲^=𝐲^𝐳^=𝐳^ | 𝐱^=xρ^−yφ^x2+y2𝐲^=yρ^+xφ^x2+y2𝐳^=𝐳^ | 𝐱^=x(x2+y2𝐫^+zθ^)−yx2+y2+z2φ^x2+y2x2+y2+z2𝐲^=y(x2+y2𝐫^+zθ^)+xx2+y2+z2φ^x2+y2x2+y2+z2𝐳^=z𝐫^−x2+y2θ^x2+y2+z2
Cylindrical  | ρ^=cos⁡φ𝐱^+sin⁡φ𝐲^φ^=−sin⁡φ𝐱^+cos⁡φ𝐲^𝐳^=𝐳^ | ρ^=ρ^φ^=φ^𝐳^=𝐳^ | ρ^=ρ𝐫^+zθ^ρ2+z2φ^=φ^𝐳^=z𝐫^−ρθ^ρ2+z2
Spherical  | 𝐫^=sin⁡θ(cos⁡φ𝐱^+sin⁡φ𝐲^)+cos⁡θ𝐳^θ^=cos⁡θ(cos⁡φ𝐱^+sin⁡φ𝐲^)−sin⁡θ𝐳^φ^=−sin⁡φ𝐱^+cos⁡φ𝐲^ | 𝐫^=sin⁡θρ^+cos⁡θ𝐳^θ^=cos⁡θρ^−sin⁡θ𝐳^φ^=φ^ | 𝐫^=𝐫^θ^=θ^φ^=φ^

## 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 angleα

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Vector field A | Ax𝐱^+Ay𝐲^+Az𝐳^ | Aρρ^+Aφφ^+Az𝐳^ | Ar𝐫^+Aθθ^+Aφφ^
Gradient ∇f[1] | ∂f∂x𝐱^+∂f∂y𝐲^+∂f∂z𝐳^ | ∂f∂ρρ^+1ρ∂f∂φφ^+∂f∂z𝐳^ | ∂f∂r𝐫^+1r∂f∂θθ^+1rsin⁡θ∂f∂φφ^
Divergence ∇ ⋅ A[1] | ∂Ax∂x+∂Ay∂y+∂Az∂z | 1ρ∂(ρAρ)∂ρ+1ρ∂Aφ∂φ+∂Az∂z | 1r2∂(r2Ar)∂r+1rsin⁡θ∂∂θ(Aθsin⁡θ)+1rsin⁡θ∂Aφ∂φ
Curl ∇ × A[1] | (∂Az∂y−∂Ay∂z)𝐱^+(∂Ax∂z−∂Az∂x)𝐲^+(∂Ay∂x−∂Ax∂y)𝐳^ | (1ρ∂Az∂φ−∂Aφ∂z)ρ^+(∂Aρ∂z−∂Az∂ρ)φ^+1ρ(∂(ρAφ)∂ρ−∂Aρ∂φ)𝐳^ | 1rsin⁡θ(∂∂θ(Aφsin⁡θ)−∂Aθ∂φ)𝐫^+1r(1sin⁡θ∂Ar∂φ−∂∂r(rAφ))θ^+1r(∂∂r(rAθ)−∂Ar∂θ)φ^
Laplace operator ∇2f ≡ ∆f[1] | ∂2f∂x2+∂2f∂y2+∂2f∂z2 | 1ρ∂∂ρ(ρ∂f∂ρ)+1ρ2∂2f∂φ2+∂2f∂z2 | 1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin2θ∂2f∂φ2
Vector gradient ∇Aβ | ∂Ax∂x𝐱^⊗𝐱^+∂Ax∂y𝐱^⊗𝐲^+∂Ax∂z𝐱^⊗𝐳^+∂Ay∂x𝐲^⊗𝐱^+∂Ay∂y𝐲^⊗𝐲^+∂Ay∂z𝐲^⊗𝐳^+∂Az∂x𝐳^⊗𝐱^+∂Az∂y𝐳^⊗𝐲^+∂Az∂z𝐳^⊗𝐳^ | ∂Aρ∂ρρ^⊗ρ^+(1ρ∂Aρ∂φ−Aφρ)ρ^⊗φ^+∂Aρ∂zρ^⊗𝐳^+∂Aφ∂ρφ^⊗ρ^+(1ρ∂Aφ∂φ+Aρρ)φ^⊗φ^+∂Aφ∂zφ^⊗𝐳^+∂Az∂ρ𝐳^⊗ρ^+1ρ∂Az∂φ𝐳^⊗φ^+∂Az∂z𝐳^⊗𝐳^ | ∂Ar∂r𝐫^⊗𝐫^+(1r∂Ar∂θ−Aθr)𝐫^⊗θ^+(1rsin⁡θ∂Ar∂φ−Aφr)𝐫^⊗φ^+∂Aθ∂rθ^⊗𝐫^+(1r∂Aθ∂θ+Arr)θ^⊗θ^+(1rsin⁡θ∂Aθ∂φ−cot⁡θAφr)θ^⊗φ^+∂Aφ∂rφ^⊗𝐫^+1r∂Aφ∂θφ^⊗θ^+(1rsin⁡θ∂Aφ∂φ+cot⁡θAθr+Arr)φ^⊗φ^
Vector Laplacian ∇2A ≡ ∆A[2] | ∇2Ax𝐱^+∇2Ay𝐲^+∇2Az𝐳^ |

(∇2Aρ−Aρρ2−2ρ2∂Aφ∂φ)ρ^+(∇2Aφ−Aφρ2+2ρ2∂Aρ∂φ)φ^+∇2Az𝐳^

(∇2Ar−2Arr2−2r2sin⁡θ∂(Aθsin⁡θ)∂θ−2r2sin⁡θ∂Aφ∂φ)𝐫^+(∇2Aθ−Aθr2sin2θ+2r2∂Ar∂θ−2cos⁡θr2sin2θ∂Aφ∂φ)θ^+(∇2Aφ−Aφr2sin2θ+2r2sin⁡θ∂Ar∂φ+2cos⁡θr2sin2θ∂Aθ∂φ)φ^

Directional derivative (A ⋅ ∇)B[3] | 𝐀⋅∇Bx𝐱^+𝐀⋅∇By𝐲^+𝐀⋅∇Bz𝐳^ | (Aρ∂Bρ∂ρ+Aφρ∂Bρ∂φ+Az∂Bρ∂z−AφBφρ)ρ^+(Aρ∂Bφ∂ρ+Aφρ∂Bφ∂φ+Az∂Bφ∂z+AφBρρ)φ^+(Aρ∂Bz∂ρ+Aφρ∂Bz∂φ+Az∂Bz∂z)𝐳^ |

(Ar∂Br∂r+Aθr∂Br∂θ+Aφrsin⁡θ∂Br∂φ−AθBθ+AφBφr)𝐫^+(Ar∂Bθ∂r+Aθr∂Bθ∂θ+Aφrsin⁡θ∂Bθ∂φ+AθBrr−AφBφcot⁡θr)θ^+(Ar∂Bφ∂r+Aθr∂Bφ∂θ+Aφrsin⁡θ∂Bφ∂φ+AφBrr+AφBθcot⁡θr)φ^

Tensor divergence ∇ ⋅ Tγ |

(∂Txx∂x+∂Tyx∂y+∂Tzx∂z)𝐱^+(∂Txy∂x+∂Tyy∂y+∂Tzy∂z)𝐲^+(∂Txz∂x+∂Tyz∂y+∂Tzz∂z)𝐳^

[∂Tρρ∂ρ+1ρ∂Tφρ∂φ+∂Tzρ∂z+1ρ(Tρρ−Tφφ)]ρ^+[∂Tρφ∂ρ+1ρ∂Tφφ∂φ+∂Tzφ∂z+1ρ(Tρφ+Tφρ)]φ^+[∂Tρz∂ρ+1ρ∂Tφz∂φ+∂Tzz∂z+Tρzρ]𝐳^

[∂Trr∂r+2Trrr+1r∂Tθr∂θ+cot⁡θrTθr+1rsin⁡θ∂Tφr∂φ−1r(Tθθ+Tφφ)]𝐫^+[∂Trθ∂r+2Trθr+1r∂Tθθ∂θ+cot⁡θrTθθ+1rsin⁡θ∂Tφθ∂φ+Tθrr−cot⁡θrTφφ]θ^+[∂Trφ∂r+2Trφr+1r∂Tθφ∂θ+1rsin⁡θ∂Tφφ∂φ+Tφrr+cot⁡θr(Tθφ+Tφθ)]φ^

* ^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 ∂i𝐀⊗𝐞i. An alternative definition is 𝐞i⊗∂i𝐀.
* ^γ Defined in Cartesian coordinates as 𝐞i⋅∂i𝐓. An alternative definition is ∂i𝐓⋅𝐞i.

### Differential elements

Operation  | Cartesian coordinates
(x, y, z) | Cylindrical coordinates
(ρ, φ, z) | Spherical coordinatesα
(r, θ, φ)

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Differential displacement dℓ[1] | dx𝐱^+dy𝐲^+dz𝐳^ | dρρ^+ρdφφ^+dz𝐳^ | dr𝐫^+rdθθ^+rsin⁡θdφφ^
Differential normal area dS | dydz𝐱^+dxdz𝐲^+dxdy𝐳^ | ρdφdzρ^+dρdzφ^+ρdρdφ𝐳^ | r2sin⁡θdθdφ𝐫^+rsin⁡θdrdφθ^+rdrdθφ^
Differential volume dV[1] | dxdydz | ρdρdφdz | r2sin⁡θdrdθdφ

### Calculation rules

1. div⁡grad⁡f≡∇⋅∇f≡∇2f
2. curl⁡grad⁡f≡∇×∇f=𝟎
3. div⁡curl⁡𝐀≡∇⋅(∇×𝐀)=0
4. curl⁡curl⁡𝐀≡∇×(∇×𝐀)=∇(∇⋅𝐀)−∇2𝐀 (Lagrange's formula for del)
5. ∇2(fg)=f∇2g+2∇f⋅∇g+g∇2f
6. ∇2(𝐏⋅𝐐)=𝐐⋅∇2𝐏−𝐏⋅∇2𝐐+2∇⋅[(𝐏⋅∇)𝐐+𝐏×∇×𝐐] (From [4] )

## Cartesian derivation

File:Nabla cartesian.svg

div⁡𝐀=limV->0∬∂V𝐀⋅d𝐒∭VdV=[Ax(x+dx)−Ax(x)]dydz+[Ay(y+dy)−Ay(y)]dxdz+[Az(z+dz)−Az(z)]dxdydxdydz=∂Ax∂x+∂Ay∂y+∂Az∂z

(curl⁡𝐀)x=limS⊥x^->0∫∂S𝐀⋅dℓ∬SdS=[Az(y+dy)−Az(y)]dz−[Ay(z+dz)−Ay(z)]dydydz=∂Az∂y−∂Ay∂z

The expressions for (curl⁡𝐀)y and (curl⁡𝐀)z are found in the same way.

## Cylindrical derivation

File:Nabla cylindrical2.svg

div⁡𝐀=limV->0∬∂V𝐀⋅d𝐒∭VdV=[Aρ(ρ+dρ)(ρ+dρ)−Aρ(ρ)ρ]dϕdz+[Aϕ(ϕ+dϕ)−Aϕ(ϕ)]dρdz+[Az(z+dz)−Az(z)]dρ(ρ+dρ/2)dϕρdϕdρdz=1ρ∂(ρAρ)∂ρ+1ρ∂Aϕ∂ϕ+∂Az∂z

(curl⁡𝐀)ρ=limS⊥ρ^->0∫∂S𝐀⋅dℓ∬SdS=Aϕ(z)(ρ+dρ)dϕ−Aϕ(z+dz)(ρ+dρ)dϕ+Az(ϕ+dϕ)dz−Az(ϕ)dz(ρ+dρ)dϕdz=−∂Aϕ∂z+1ρ∂Az∂ϕ

(curl⁡𝐀)ϕ=limS⊥ϕ^->0∫∂S𝐀⋅dℓ∬SdS=Az(ρ)dz−Az(ρ+dρ)dz+Aρ(z+dz)dρ−Aρ(z)dρdρdz=−∂Az∂ρ+∂Aρ∂z

(curl⁡𝐀)z=limS⊥z^->0∫∂S𝐀⋅dℓ∬SdS=Aρ(ϕ)dρ−Aρ(ϕ+dϕ)dρ+Aϕ(ρ+dρ)(ρ+dρ)dϕ−Aϕ(ρ)ρdϕρdρdϕ=−1ρ∂Aρ∂ϕ+1ρ∂(ρAϕ)∂ρ

curl⁡𝐀=(curl⁡𝐀)ρρ^+(curl⁡𝐀)ϕϕ^+(curl⁡𝐀)zz^=(1ρ∂Az∂ϕ−∂Aϕ∂z)ρ^+(∂Aρ∂z−∂Az∂ρ)ϕ^+1ρ(∂(ρAϕ)∂ρ−∂Aρ∂ϕ)z^

## Spherical derivation

File:Nabla spherical2.svg div⁡𝐀=limV->0∬∂V𝐀⋅d𝐒∭VdV=[Ar(r+dr)(r+dr)2−Ar(r)r2]sin⁡θdθdϕ+[Aθ(θ+dθ)sin⁡(θ+dθ)−Aθ(θ)sin⁡θ]rdrdϕ+[Aϕ(ϕ+dϕ)−Aϕ(ϕ)]rdrdθdrrdθrsin⁡θdϕ=1r2∂(r2Ar)∂r+1rsin⁡θ∂(Aθsin⁡θ)∂θ+1rsin⁡θ∂Aϕ∂ϕ

(curl⁡𝐀)r=limS⊥r^->0∫∂S𝐀⋅dℓ∬SdS=Aθ(ϕ)rdθ+Aϕ(θ+dθ)rsin⁡(θ+dθ)dϕ−Aθ(ϕ+dϕ)rdθ−Aϕ(θ)rsin⁡(θ)dϕrdθrsin⁡θdϕ=1rsin⁡θ∂(Aϕsin⁡θ)∂θ−1rsin⁡θ∂Aθ∂ϕ

(curl⁡𝐀)θ=limS⊥θ^->0∫∂S𝐀⋅dℓ∬SdS=Aϕ(r)rsin⁡θdϕ+Ar(ϕ+dϕ)dr−Aϕ(r+dr)(r+dr)sin⁡θdϕ−Ar(ϕ)drdrrsin⁡θdϕ=1rsin⁡θ∂Ar∂ϕ−1r∂(rAϕ)∂r

(curl⁡𝐀)ϕ=limS⊥ϕ^->0∫∂S𝐀⋅dℓ∬SdS=Ar(θ)dr+Aθ(r+dr)(r+dr)dθ−Ar(θ+dθ)dr−Aθ(r)rdθrdrdθ=1r∂(rAθ)∂r−1r∂Ar∂θ

curl⁡𝐀=(curl⁡𝐀)rr^+(curl⁡𝐀)θθ^+(curl⁡𝐀)ϕϕ^=1rsin⁡θ(∂(Aϕsin⁡θ)∂θ−∂Aθ∂ϕ)r^+1r(1sin⁡θ∂Ar∂ϕ−∂(rAϕ)∂r)θ^+1r(∂(rAθ)∂r−∂Ar∂θ)ϕ^

## 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, ∂𝐫∂u=∂s∂u𝐮 where s is the arc length parameter.

For two sets of coordinate systems ui and vj, according to chain rule, d𝐫=∑i∂𝐫∂uidui=∑i∂s∂ui𝐮^idui=∑j∂s∂vj𝐯^jdvj=∑j∂s∂vj𝐯^j∑i∂vj∂uidui=∑i∑j∂s∂vj∂vj∂ui𝐯^jdui.

Now, we isolate the ith component. For i≠k, let duk=0. Then divide on both sides by dui to get: ∂s∂ui𝐮^i=∑j∂s∂vj∂vj∂ui𝐯^j.

## See also

* Del
* Orthogonal coordinates
* Curvilinear coordinates
* Vector fields in cylindrical and spherical coordinates

## References

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.
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.

## External links

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

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