  In this paper we study symmetries, Newtonoid vector fields, conservation
laws, Noether's Theorem and its converse, in the framework of the
$k$-symplectic formalism, using the Fr\"olicher-Nijenhuis formalism on the
space of $k^1$-velocities of the configuration manifold.
  For the case $k=1$, it is well known that Cartan symmetries induce and are
induced by constants of motions, and these results are known as Noether's
Theorem and its converse. For the case $k>1$, we provide a new proof for
Noether's Theorem, which shows that, in the $k$-symplectic formalism, each
Cartan symmetry induces a conservation law. We prove that, under some
assumptions, the converse of Noether's Theorem is also true and we provide
examples when this is not the case. We also study the relations between
dynamical symmetries, Newtonoid vector fields, Cartan symmetries and
conservation laws, showing when one of them will imply the others. We use
several examples of partial differential equations to illustrate when these
concepts are related and when they are not.
