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.