The next technique that Cartan employed in his book on Lecons on Integraux
Invariants is the concept of the exterior derivative (now called the exterior
differential). 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 an element of tensor type (n,0) into an element of
tensor type (n,1). What is remarkable is that the operational definition of the exterior derivative works on any pform 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 N1 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 3component vector from (n,1) into another 3component (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.
