Exterior form

of degree \$ r \$, exterior \$ r \$- form

A homogeneous element of degree \$ r \$ of the exterior algebra \$ \wedge V \$ of a vector space \$ V \$, i.e. an element of the \$ r \$- th exterior power \$ \wedge ^ {r} V \$. The expression "exterior form of degree r on the space V" usually denotes a skew-symmetric \$ r \$- linear function (or a skew-symmetric \$ r \$ times covariant tensor) on \$ V \$. The direct sum of the spaces of skew-symmetric \$ r \$- linear functions on \$ V \$, \$ r = 0, 1 \dots \$ endowed with the exterior product, is an algebra isomorphic to the exterior algebra \$ \wedge V ^ {*} \$.

Under an exterior form one also understands a differential form.

How to Cite This Entry:
Exterior form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exterior_form&oldid=46888
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article