This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Notes

Coordinate conversions

Conversion between Cartesian, cylindrical, and spherical coordinates[1]
From
Cartesian Cylindrical Spherical
To Cartesian
Cylindrical
Spherical

CAUTION: the operation 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
Cylindrical
Spherical
Conversion between unit vectors in Cartesian, cylindrical, and spherical coordinate systems in terms of source coordinates
Cartesian Cylindrical Spherical
Cartesian
Cylindrical
Spherical

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, φ, θ),
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 gradient A
Vector Laplacian 2A ≡ ∆A[2]

Directional derivativeα[3] (A ⋅ ∇)B

Tensor divergence ∇ ⋅ T

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.

Calculation rules

  1. (Lagrange's formula for del)
  2. (From [4] )

Cartesian derivation

The expressions for and are found in the same way.

Cylindrical derivation

Spherical derivation

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