  The ``time-evolution operator'' in mechanics is a powerful tool which can be
geometrically defined as a vector field along the Legendre map. It has been
extensively used by several authors for studying the structure and properties
of the dynamical systems (mainly the non-regular ones), such as the relation
between the Lagrangian and Hamiltonian formalisms, constraints, and
higher-order mechanics. This paper is devoted to defining a generalization of
this operator for field theories, in a covariant formulation. In order to do
this, we also use sections along maps, in particular multivector fields
(skew-symmetric contravariant tensor fields of order greater than 1), jet
fields and connection forms along the Legendre map. As a first relevant
property, we use these geometrical objects to obtain the solutions of the
Lagrangian and Hamiltonian field equations, and the equivalence among them
(specially for non-regular field theories).
