Cartan utilized two new concepts in his study of Integral Invariants. He
devised an algebra based upon the concept of exterior multiplication, an idea to
be found in the earlier works of Grassman. The exterior algebra is not the
algebra of real numbers that are learned in the elementary grades. For certain
(odd dimensional) elements of the algebra, A times B is equal to the negative of
B times A. This construction will be written as A^B=B^A.
Until the last few years, only specialists knew how to utilize and apply the
concepts. However, in the engineering world, the idea shows up through the
Gibbs cross product in vector analysis, where AxB =
BxA. (Not the same as A^B = B^A, as the exterior product is
associative, but the Gibbs cross product is not.) Unfortunately, the ubiquitous
Gibbs cross product is not useful
The Cartan method of exterior algebra develops the antisymmetric concepts
associated with the exterior product, A^B=B^A, and the auxiliary result A^A=0,
(A=odd) into what is called a graded algebra. Every element of the Algebra is
composed of primitive basis elements of an N dimensional linear vector space,
and the 1 dimensional space of functions. Through the exterior (wedge ^)
product higher dimensional composites may be constructed, forming (n,p)
components of higher order linear structures. The algebraic process can be
continued until a total of 2^N components are constructed. The algebraic
exterior vector space has 2^N components. For example, for N=4, the 1 dimensional subspace of functions called 0forms, and the N dimensional space of 1forms, can be multiplied to construct the (n,2)= 6 dimensional space of 2forms, the (n,3)=4 dimensional space of 3 forms, and the (n,4)=1 dimensional space of 4forms. Note the correspondence to the Pascal triangle. Higher order algebraic composites are null. The algebra is said to be closed. This closure feature is a topological idea.
In physics, these exterior algebra subspaces often are called the scalars,
vectors, tensors, pseudo_vectors, and the pseudo_scalars. The next thing that
Cartan employed is the concept of the exterior derivative, which is a rule of
differentiation that transports an element of a lower dimensional exterior
algebra subspace to the next higher dimensional subspace. The primitive example
is the use of the gradient operator acting on a function to construct a gradient
vector field.
