  We derive the variational principle and Noether's theorem in generally
covariant field theory in an explicitly coordinate-independent way by means of
the exterior calculus over the space-time manifold. We then focus on the
symmetry of active diffeomorphisms, that is, the pushforwards along the
integral lines of any vector field, and its analogies with internal gauge
symmetries. For instance, it is well known that a class of Noether currents
associated to a gauge symmetry can be obtained by taking the partial derivative
of the Lagrangian with respect to the corresponding gauge field. Here we show
that this relation also holds for the Noether currents associated to
diffeomorphisms and the vielbein, but only if one decomposes all forms in the
vielbein basis. We also relate the diffeomorphism Noether currents to the
matter energy-momentum tensor of General Relativity, to Hamiltonian boundary
terms and to two known energy-momentum complexes of the vielbein.
