Exterior algebra

In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors and , denoted by is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of can be interpreted as the area of the parallelogram with sides and which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that for all vectors and but, unlike the cross product, the exterior product is associative (after introducing the exterior cubic, that is, oriented volume). When regarded in this manner, the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number of vectors can be defined and is sometimes called a -blade (or decomposable, or simple, by some authors). It lives in a space known as the -th exterior power (generalizing exterior square and exterior cubic). Blades are basic objects in Projective Geometry, where no measure for length or angle (hence no parallelism) is assumed, but the main structure in there is linearity. If Euclidean product is given for the vectors, the magnitude (that is, a scalar) of the resulting -blade is the oriented hypervolume of the -dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors. The exterior algebra provides an algebraic setting to answer some type of geometric questions.