Cartan's Calculus: the interior product.

None of the last two concepts, the concept of wedge exterior product, and the concept of the exterior derivative depend upon the notion of a metric space. The exterior operations are defined on a variety for which the constraints of metric have not been defined. The exterior derivative is also independent of a connection. It is not equivalent to the tensor Covariant derivative except under special circumstances. Later on, the concept of a Covariant derivative will be explored in detail. The primitive idea is that a Covariant derivative is a constrained or restricted process which when acting on a tensor produces another tensor.

Another operation in the Cartan exterior algebra is useful. It is the concept of the interior product. The interior product is essentially the concept of projective transversality or collinearity. The interior product operation may look like the scalar product of ordinary vector analysis, but the idea is a bit more complex. It is best to think of the interior product as an operation that linearly takes a p element into a p-1 element of the exterior algebra. Given a p form on the final state, and operation that takes a p form to a p-1 form on the final state, it must follow that the pullback of the p-1 form is well defined. However, a contravariant vector field on the final state is not well defined on the initial state if the inverse Jacobian matrix does not exist. However a contravaraint vector density on the final state can be pulled back to the initial state by means of the adjoint of the Jacobian matrix, even though the inverse is not defined. Hence, the inner product of a p-form and a contravariant vector density pulled back is well defined in terms of the interior product of the pullback of the p-form and the pull back of the contravariant vector density.

The interior product of a contravariant vector density or current with a 1-form creates a 0-form, or function. The interior product is often utilized in terms of a contravariant vectors, not contravariant vector densities, but then the result is only well defined with respect to diffeomorphisms, for which the inverse Jacobian is defined.

It is best to think of the interior product as being with respect to "currents", and not with respect to vectors. Currents are N-1 form structures on the N dimensional exterior algebra dual to the 1-form. Currents on the final state are well defined on the initial state with respect to the processes of adjoint pullback and functional substitution. The process is dual to the transpose pullback and functional substitution of p-forms.

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