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 can be used instead of the mathematical function owing to its domain and image. The classical arctan function has an image of , whereas atan2 is defined to have an image of .
Coordinate conversions
+ Conversion between Cartesian, cylindrical, and spherical coordinates From Cartesian Cylindrical SphericalTo Cartesian Cylindrical Spherical Note that 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 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 Cylindrical coordinates Spherical coordinates ,where is the polar angle and is the azimuthal angle Vector field Gradient Divergence Curl Laplace operator Vector gradient Vector Laplacian Directional derivative Tensor divergence
Differential elements Operation Cartesian coordinates Cylindrical coordinates Spherical coordinates Differential displacement Differential normal area Differential volume
Calculation rules
(Lagrange's formula for del)
(From )
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 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 Del Orthogonal coordinates Curvilinear coordinates Vector fields in cylindrical and spherical coordinates
References
External links Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates.