Improper currents in theories with local invariance

We present proofs of two results: (a) 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; (b) When there is a current which is covariantly conserved, it differs from the canonical current by an improper current. Both proofs are 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. Both proofs are made using only . These are alternative proofs of known results which, besides making use of elementary calculus only, rendering them accessible to a large number of physicists, present the novelty of providing, in both cases, explicit formulae for the decomposition of the improper currents into their two terms (the one vanishing on-shell and the one whose divergence vanishes off-shell).