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Del in cylindrical and spherical coordinates

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

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

Unit vector conversions

Del formula

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Differential elements

Calculation rules

  4. (Lagrange's formula for del)
  6. (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 <sup>th</sup> component. For , let . Then divide on both sides by to get:

See also

References

External links