Area Element

The area element for a Surface with Riemannian Metric

$$ds^2=E\,du^2+2F\,du\,dv+G\,dv^2$$

is

$$dA=\sqrt{EG-F^2}\,du\wedge dv,$$

where $du\wedge dv$ is the Wedge Product.

See also Area, Line Element, Riemannian Metric, Volume Element

References

Gray, A. ``The Intuitive Idea of Area on a Surface.'' §13.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 259-260, 1993.