Cartan's theory of exterior differential forms is NOT just another notational system of fancy. The importance of the Cartan concept resides with the fact that differential forms are well defined objects with respect to Functional Substitution and the PullBack (FS_PB for short) relative to C1 differentiable maps from an initial variety of variables to a final state or variety of variables. The evolutionary map does not need to have an inverse (or an invertible Jacobian) and yet the forms and their behavior are well defined in functional format with regards to FS_PB. A similar statement cannot be made about other types of tensor structures, in general.

As P. Libermann points out, higher order differential processes (jets) on the tangent space (of vector like velocity and acceleration) are not always linear (hence do not form a vector bundle), but on the co-tangent space (of differential forms) everything is well behaved. These are the concepts that break the notion of wave (cotangent) particle (tangent) duality, so cherished in Copenhagen quantum mechanics.

The point is that without metric, without the constraint of a group based connection, without restriction to diffeomorphisms or even homeomorphisms, differential forms are well defined quantities in a FS_PB sense. They can be used to describe situations that involve topological evolution, while the usual contra-variant tensor fields of particle mechanics are restricted (usually) to reversible evolution that preserves topological features.

The coefficients of differential forms are either anti-symmetric co-variant tensors, or tensor densities, the most useful of field structures used to describe physical systems. To appreciate these extraordinary features of differential forms, consider C1 maps from a space (variety) of M dimensions to a space of N dimensions. Construct a differential form on (meaning in terms of the variables of) the final state, and then by functional substitution and use of the Jacobian map construct the well defined functional form of the tensor coefficients of the differential form on the initial state. This is what is meant by Retrodiction. The process works even though the inverse map and/or inverse Jacobian does not exist. Remember the Jacobian is a N by M matrix and is without an inverse.

Note that the Retrodiction process fails for Contravariant tensors! Moreover, the complimentary process of Prediction always fails for any type of tensor field when the inverse map and/or inverse Jacobian does not exist. (Prediction would be defined as functional substitution and push forward).

The bottom line is that differential forms carry topological content, and can be used to study irreversible phenomena, in a deterministic, but retrodictive, manner.

Tensors, Jacobians and Metrics

Differential forms Notes 1

Classical Field theory Notes 1

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