[1] Differential invariants were introduced in special cases by Sophus Lie in the early 1880s and studied by Georges Henri Halphen at the same time.
The moving frames method, which is a refinement of Élie Cartan's method of moving frames, gives several new powerful tools for finding and classifying the equivalence and symmetry properties of submanifolds, differential invariants, and their syzygies.
Then G also acts, locally, on the space of all graphs of the form y = ƒ(x).
The action of G on the first derivative, for instance, is such that the chain rule continues to hold: if then Similar considerations apply for the computation of higher prolongations.
The group G acts, locally, on the space of such graphs, and induces an action on the k-th prolongation Y(k) consisting of graphs passing through each point modulo the relation of k-th order contact.