  This paper (the seventh paper in a series of eight) continues the development
of our theory of multivector and extensor calculus on smooth manifolds. Here we
deal first with the concepts of ordinary Hodge coderivatives, duality
identities, and Hodge coderivative identities. Then, we recall the concept of a
Levi-Civita geometric structure and the concepts of Levi-Civita and gauge
derivatives. New formulas that are important in the Lagrangian theory of
multivector adn extensor fields are obtained. We introduce also he concept of
covariant Hodge coderivative. We detail how all these concepts are related.
