Cartan's Calculus: The exterior product

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
except in dimension 3, and as mentioned above, the algebra so constructed is not associative. It is a pity that the Gibbs formalism (which replaced the Hamiltonian quaterion formalism) has been so ingrained into our educational system.

The Cartan method of exterior algebra develops the anti-symmetric 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 0-forms, and the N dimensional space of 1-forms, can be multiplied to construct the (n,2)= 6 dimensional space of 2-forms, the (n,3)=4 dimensional space of 3 forms, and the (n,4)=1 dimensional space of 4-forms. 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.

 As I have time I will replace this under construction symbol with a link to a more detailed pdf or tex file download.