A Gauss-Ostrogradskii theorem for integral transforms over the Euler characteristic - To the memory of professor Anatolii Platonovich Prudnikov

We prove an analog of the Gauss-Ostrogradskii theorem for integration over the Euler characteristic, expressing the Euler characteristic of a manifold with a boundary in terms of the zeros of a smooth dynamical system and its behaviour on the boundary. This result makes it possible to compute the Euler characteristic of a closed manifold via behaviour of a discontinuous dynamical system. The Gauss-Ostrogradskii theorem for the Euler characteristic also clarifies certain classial computations.