In this paper we give a differential lifting principle which
provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The
topological fields we consider here are in fact topological systems in
the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric
alternative to the axiomatizations obtained by Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for
some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving
that, in existentially closed models of a large theory of differential
topological fields, the jet-spaces are dense in their ambient topological space.