Blade (geometry)

In the study of geometric algebras, a blade is a generalization of the concept of scalars and vectors to include "simple" bivectors, trivectors, etc. Specifically, a -blade is any object that can be expressed as the exterior product (informally "wedge product") of vectors, and is of "grade" .
In detail:
For an -dimensional space, there are blades of all grades from 0 to inclusive. A vector subspace of finite dimension may be represented by the -blade formed as a wedge product of all the elements of a basis for that subspace. Indeed, a -blade is naturally equivalent to a -subspace endowed with a volume form (an alternating -multilinear scalar-valued function) normalized to take unit value on the -blade.
Examples.
In two-dimensional space, scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades known as pseudoscalars, in that they are elements of a one-dimensional space distinct from regular scalars.
In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, while 2-blades are oriented area elements. Here 3-blades represent three-dimensional volume elements, which form a one-dimensional vector space similar to scalars. Unlike scalars, 3-blades transform according to the Jacobian determinant of a change-of-coordinate function.