Del in cylindrical and spherical coordinates


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

Coordinate conversions

Note that the operation must be interpreted as the two-argument inverse tangent, atan2.

Unit vector conversions

CartesianCylindricalSpherical
Cartesian
Cylindrical
Spherical

CartesianCylindricalSpherical
Cartesian
Cylindrical
Spherical

Del formula


OperationCartesian 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


Calculation rules

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Cartesian derivation

The expressions for and are found in the same way.

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: