Del in cylindrical and spherical coordinates
This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.
Coordinate conversions
Unit vector conversions
| Cartesian | Cylindrical | Spherical | |
| Cartesian | |||
| Cylindrical | |||
| Spherical |
| Cartesian | Cylindrical | Spherical | |
| Cartesian | |||
| Cylindrical | |||
| Spherical |
Del formula
| Operation | Cartesian coordinates | Cylindrical coordinates | Spherical coordinates, where φ is the azimuthal and is the polar angle |
| Vector field | |||
| Gradient | |||
| Divergence | |||
| Curl | |||
| Laplace operator | |||
| Vector Laplacian | |||
| Material derivative | |||
| Tensor | |||
| Differential displacement | |||
| Differential normal area | |||
| Differential volume |
Non-trivial calculation rules
Cartesian derivation
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 component. For, let. Then divide on both sides by to get: