The formal theory of partial differential equations is a geometric approach to systems of PDEs using the formalism of jet bundles. The n-th order jet bundle is the bundle over the space of independent and dependent variables whose fibre coordinates are derivatives of the dependent variables of order up to n. The jet bundles of different orders are connected by the prolongation (or jet extension) and projection operations. A system of PDEs is then a fibred submanifold of a jet bundle and a solution a section of the base space whose prolongation lies within that submanifold. The key applications of this formalism are the completion of systems of PDEs to equivalent involutive systems which contain all their integrability conditions (and which thus possess formal power series solutions that can be constructed order by order) and the analysis of the continuous symmetries of a system.

rcdb100@mrao.cam.ac.uk