  A general method using multipliers for finding the conserved integrals for
any system of partial differential equations (PDEs) is reviewed and further
developed in several ways. Multipliers are expressions whose (summed) product
with a PDE (system) yields a local divergence identity which has the physical
meaning of a continuity equation involving a conserved density and a spatial
flux for solutions of the PDE (system). On spatial domains, the integral form
of a continuity equation yields a conserved integral. When a PDE (system) is
variational, multipliers correspond to symmetries of the variational principle,
and the local divergence identity relating a multiplier to a conserved integral
is the same as the variational identity used in Noether's theorem for
connecting conserved integrals to invariance of a variational principle. From
this viewpoint, the multiplier method is shown to constitute a modern form of
Noether's theorem in which the variational principle is not used. When a PDE
(system) is non-variational, multipliers are shown to be an adjoint counterpart
to symmetries, and the local divergence identity relating a multiplier to a
conserved integral is shown to be an adjoint generalization of the variational
identity underling Noether's theorem. Two main results are established for PDE
systems having a solved-form for leading derivatives, which encompasses all
typical PDEs of physical interest. First, all non-trivial conserved integrals
are shown to arise from non-trivial multipliers in a one-to-one manner, taking
into account certain equivalence freedoms. Second, a scaling formula based on
dimensional analysis is derived to obtain the conserved density and spatial
flux in any conserved integral, just using the corresponding multiplier and the
given PDE system. The derivations use a few basic tools from variational
calculus, for which a self-contained formulation is provided.
