The next technique that Cartan employed in his book on Lecons on Integraux Invariants is the concept of the exterior derivative. This concept is a rule of differentiation that transports an element of a lower dimensional exterior algebra subspace to the next higher dimensional exterior algebra subspace. The primitive example is the use of the gradient operator acting on a function to construct a gradient vector field. The gradient operation takes and element of (n,0) into an element of (n,1).

What is remarkable is that the operational definition of the exterior derivative works on any p-form carrying it into a p+1 form. For example, the exterior derivative of an p=N-1 form carries it into the a p = N form. In engineering technology, one would say the the divergence of a N-1 current (vector) produces a density. IN N=3 dimensions, the Pascal triangle of alegbraic exterior algebra subspaces has dimensions, {1,3,3,1}. In engineering practice it is recognized that the differential operations defined as the "Curl" takes a 3-component vector from (n,1) into another 3-component (psuedo) vector of the space (n,2). These two vectors have the same number of components only in a space of N=3 dimensions (that is why the Gibbs cross product is useful). However the two species of vectors are never physicaly equivalent. No engineer ever adds linear momentum to angular momentum. No engineer ever adds a Magnetic field 3 component vector to an electric field 3 component vector. The concept of the gradient and the concept of the curl are treated differently in Gibbsian vector analysis.

In the Cartan exterior calculus, the operations are the same and are represented by the same exterior derivative, but the operation operates on different algebraic subspaces.

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