  A vector calculus approach for the determination of advected invariants is
presented for inviscid fluid flow. This approach describes invariants by means
of Lie dragging of scalars, vectors, and skew-tensors with respect to the fluid
velocity, which has the physical meaning of characterizing tensorial quantities
that are frozen into the flow. Several new main results are obtained. First,
simple algebraic and differential operators that can be applied recursively to
derive a complete set of invariants starting from the basic known local and
nonlocal invariants are constructed. Second, these operators are used to derive
infinite hierarchies of local and nonlocal invariants for both adiabatic fluids
and isentropic fluids that are either incompressible, or compressible with
barotropic and non-barotropic equations of state. Each hierarchy is complete in
the sense that no further invariants can be generated from the basic local and
nonlocal invariants. All of the resulting new invariants are related to Ertel's
invariant, the Ertel-Rossby invariant, and Hollmann's invariant. In particular,
for incompressible fluid flow in which the density is non-constant across
different fluid streamlines, a new variant of Ertel's invariant and several new
variants of Hollmann's invariant are derived, where the entropy gradient is
replaced by the density gradient. Third, the physical meaning of these new
invariants and the resulting conserved integrals are discussed, and their
relationship to conserved helicities and cross-helicities is described.
