Del in cylindrical and spherical coordinates

Summary

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 : 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 : 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 N/A
Cylindrical N/A
Spherical N/A

Unit vector conversions

Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of destination coordinates[1]
Cartesian Cylindrical Spherical
Cartesian N/A
Cylindrical N/A
Spherical N/A
Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of source coordinates
Cartesian Cylindrical Spherical
Cartesian N/A
Cylindrical N/A
Spherical N/A

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
Vector field A
Gradient f[1]
Divergence ∇ ⋅ A[1]
Curl ∇ × A[1]
Laplace operator 2f ≡ ∆f[1]
Vector Laplacian 2A ≡ ∆A[2]

Material derivativeα[3] (A ⋅ ∇)B

Tensor ∇ ⋅ T (not to be confused with 2nd order tensor divergence)

Differential displacement d[1]
Differential normal area dS
Differential volume dV[1]
This page uses for the polar angle and for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses for the azimuthal angle and for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch and in the formulae shown in the table above.

Non-trivial calculation rules

  1. (Lagrange's formula for del)

Cartesian derivation

Nabla cartesian.svg

The expressions for and are found in the same way.

Cylindrical derivation

Nabla cylindrical2.svg

Spherical derivation

Nabla spherical2.svg

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 to change in direction.

Therefore,

where s is the arc length parameter.

For two sets of coordinate systems and , according to chain rule,

Now, we isolate the th component. For , let . Then divide on both sides by to get:


See also

References

  1. ^ a b c d e f g h 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. Retrieved 23 March 2011.

External links

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