CALCULUS OF FORMS ALONG A MAP ADAPTED TO THE STUDY OF SECOND-ORDER DIFFERENTIAL EQUATIONS W. Sarlet, E. Mart\'{\i}nez and A. Vandecasteele Abstract. Earlier work, in which we discussed a special class of differential forms associated to a given second-order vector field $\G$, has led the roots for a much more extensive and general study of scalar and vector-valued forms along the tangent bundle projection $\t:TM\rightarrow M$ which we briefly review first. The classification of derivations of such forms requires an additional ingredient, namely a non-linear connection. Particularly important in the case of the connection coming from a second-order vector field $\G$ are the appearance of a degree zero derivation, called the dynamical covariant derivative, and of a type (1,1) tensor field along $\t$, called the Jacobi endomorphism. We further discuss the extension of this theory to time-dependent equations, for which the general set-up is the calculus of forms along the map $\pi:\R\times TM \rightarrow \R\times M$. All essential results of the autonomous case have an analogue in the time-dependent situation, but there are a number of technical complications which make it sometimes hard to decide about the best possible approach. A number of applications are presented. The most successful of these so far concerns a complete characterization of separable equations with the aid of algebraic conditions which are directly verifiable in practice. Keywords. Forms along a map, derivations, second-order equations, separability. MS classification. 58A10, 53C05, 70D05.