C++ Class Project
C++ Class Project
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.