A volume element is the differential element whose Volume Integral over some range in a given coordinate system gives the
Volume of a solid,

(1) |

(2) |

The use of the antisymmetric Wedge Product instead of the symmetric product
is a technical
refinement often omitted in informal usage. Dropping the wedges, the volume element for Curvilinear Coordinates in
is given by

(3) | |||

(4) | |||

(5) | |||

(6) | |||

(7) |

where the latter is the Jacobian and the are Scale Factors.

**References**

Gray, A. ``Isometries of Surfaces.'' §13.2 in *Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, pp. 255-258, 1993.

© 1996-9

1999-05-26