  The integrability of multivector fields in a differentiable manifold is
studied. Then, given a jet bundle $J^1E\to E\to M$, it is shown that integrable
multivector fields in $E$ are equivalent to integrable connections in the
bundle $E\to M$ (that is, integrable jet fields in $J^1E$). This result is
applied to the particular case of multivector fields in the manifold $J^1E$ and
connections in the bundle $J^1E\to M$ (that is, jet fields in the repeated jet
bundle $J^1J^1E$), in order to characterize integrable multivector fields and
connections whose integral manifolds are canonical lifting of sections. These
results allow us to set the Lagrangian evolution equations for first-order
classical field theories in three equivalent geometrical ways (in a form
similar to that in which the Lagrangian dynamical equations of non-autonomous
mechanical systems are usually given). Then, using multivector fields; we
discuss several aspects of these evolution equations (both for the regular and
singular cases); namely: the existence and non-uniqueness of solutions, the
integrability problem and Noether's theorem; giving insights into the
differences between mechanics and field theories.
