A MULTIDIMENSIONAL VARIATIONAL INTEGRAL AND ITS EXTENSIONS

Abstract

We define a variational integral in the m–dimensional Euclidean space so that the Gauss–Green theorem holds for each vector field which is everywhere differentiable (not necessary continuously). The variational integral is then extended by a transfinite sequence of improper integrals, and the Gauss–Green theorem is proved for vector fields which are differentiable only outside fairly large exceptional sets. The variational integral and its extensions are invariant with respect to a continuously differentiable change of coordinates, and hence suitable for integration on differentiable manifolds.