The Symbolic Integration of Exact PDEs

An algorithm is described which decides if a given polynomial differential expression Δ of multivariate functions is exact, i.e. whether there exists a first integral P such thatDxP = Δ for any one x of a set of n variables and to provide the integral P. A generalization is given to allow integration in the case that the exactness is prevented by terms which contain only functions of less than n independent variables.