C++ Class Project
C++ Class Project

# Exterior Algebra

Exterior Algebra also known as Grassmann Algebra is an algebra of vectors. As members of a vector space, vectors may be added and multiplied by scalers. In physics applications the scalars are usually the real or complex numbers but in principle they can be elements of any field. In order to make an algebra out of this vector space we need to know how to multiply two vectors. For this algebra we use the Grassmann product ^. It is the generalization of cross product in 3 dimensional vector algebra.
This product introduces what is known as a multivector ie an object which is a collection of vectors. V is a 1-vector, v^w is a 2-vector, and V^W^U is a 3-vector. The Grassmann algebra is the direct sum of these multivectors up to multivectors of the dimension of the vector space. Example: (scalar,1-vector,2-vector,...,n-vector) where a 0-vector is defined to be an element of the field. It may be shown that the dimension the Grassmann algebra is two to the nth power where n is the dimension of the vector space. Therefore we have 2 to the n basis elements. If we know the composition law for these then we know how to multiply in the algebra. The law is b1^b2 = ((-1)**pq)b2^b1 where p,q are the degrees of the multivectors b1,b2. Enough math, for further info on this algebra I refer you to "Differential Geometry, Gauge Theories, and Gravity" by Gockeler & Schucker.
There are an enormous amount of applications of the formalism. In physics we have the differential forms which are elements of the Grassmann Algebra over the dual vector space V*. These are used in everything from E&M (for which the provide a quite convenient calculational environment) to Hamiltonian Mechanics and Field Theory. In mathematics they lead to the concept of exterior differential systems which contain all differential and partial differential equations, sets of these, and some further problems which are unexpressable as DE's or PDE's. This formulation of these equations leads to important methods in nonlinear problems and thus promises to be quite important to all of physics in general. There are many other applications which are left to the imagination.
Anyway this is not intended to be a complete introduction to Grassmann albebra but only brief overview to allow the reader some sence as to what is being calculated in the following program as well as to the general importance of this class in scientific and mathematical applications.