  A set of differential operators acting by continuous deformations on path
dependent functionals of open and closed curves is introduced. Geometrically,
these path operators are interpreted as infinitesimal generators of curves in
the base manifold of the gauge theory. They furnish a representation with the
action of the group of loops having a fundamental role. We show that the path
derivative, which is covariant by construction, satisfies the Ricci and Bianchi
identities. Also, we provide a geometrical derivation of covariant Taylor
expansions based on particular deformations of open curves. The formalism
includes, as special cases, other path dependent operators such as end point
derivatives and area derivatives.
