  The objective of the present paper (the second in a series of four) is to
give a theory of multivector and extensor fields on a smooth manifold M of
arbitrary topology based on the powerful geometric algebra of multivectors and
extensors. Our approach does not suffer the problems of earlier attempts which
are restricted to vector manifolds. It is based on the existence of canonical
algebraic structures over the so-called canonical space associated to a local
chart (U_{o},phi_{o}) of the maximal atlas of M. The key concepts of
a-directional ordinary derivatives of multivector and extensor fields are
defined and their properties studied. Also, we introduce the Lie algebra of
smooth vector fields and the Hestenes derivatives whose properties are studied
in details.
