Del in cylindrical and spherical coordinates


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


  • 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 conversionsEdit

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

CAUTION: the operation   must be interpreted as the two-argument inverse tangent, atan2.

Unit vector conversionsEdit

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

Del formulaEdit

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 Gradient A      
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.

Calculation rulesEdit

  4.   (Lagrange's formula for del)

Cartesian derivationEdit




The expressions for   and   are found in the same way.

Cylindrical derivationEdit







Spherical derivationEdit







Unit vector conversion formulaEdit

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.


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 alsoEdit


  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 linksEdit

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