Improperness of noether currents arising from local invariance

In this paper, we present a proof that the currents arising from Noether’s first theorem in a physical theory with local invariance can always be decomposed into two terms, one of them vanishing on-shell, and the other having an off-shell vanishing divergence, or that they are improper, using the original terminology of Noether. This is a well-known result, where the novelty lies in providing an explicit formula for the terms showing up in the decomposition. The proof is performed in the most general case, that is, for arbitrary maximal order of the derivatives of the dynamical fields of the theory in the Lagrangian, and for arbitrary maximal order of the derivatives of the parameters of the symmetry transformations present in the infinitesimal transformations of the fields and spacetime coordinates. The proof is made using only elementary calculus, making it accessible to a large number of physicists.