A quasi-linear hyperbolic system of the first order, in conservative form, is considered and a supplementary conservation law is supposed to exist, as a consequence of the field equations.
Starting from a paper of K.O. Friedrichs the definition of convex covariant density is introduced and it is proven through an explicitely covariant formalism that:
a) a main field U' exists depending only on the field equations and the supplementary conservation law, but invariant through field variable mapping;
b) the system assumes a symmetric conservative form if U' is chosen as field variable and the symmetric system is generated by the knowledge of only one four-vector;
c) it is possible to define a covariant scalar function on a shock manifold which provides entropy growth (in the sense of P. D. Lax);
d) the previous function generates the shock and the shock manifolds are not space-like if the characteristic ones are not space-like.
Finally the system of relativistic fluid dynamics is shown to possess a convex covariant density and consequences of the results a-d) are discussed in detail.