Prof. Antoni Pierzchalski: The divergence theorem for vector fields

Abstract. The divergence theorem known from the calculus course and asserting - in its classic version - that the integral of the divergence of a vector field over a bordered domain is equal to the integral over the boundary of the field being a projection of this field onto the outer normal direction to boundary. The fame of theorem comes from a wide variety of its possible applications in PDE's, geometry, physics and engineering. The most staking are: the proof of the Archimedean principle on hydrostatic buoyance acting on a body immersed in a fluid or a proof of the Pythagorean theorem for multidimensional simplexes or parallelograms. The uniqueness of theorem comes from its vector character: many nontrivial but natural boundary conditions can be posed. The classic version, applications and a generalization from vector fields to vector valued forms will be discussed.