A multidimensional variational integral and its extensions
A variational integral is defined in the m-dimensional Euclidean space for which the Gauss-Green theorem holds for each vector field which is everywhere differentiable, but not necessarily continuous. An extension of this variational integral through a transfinite sequence of improper integrals is then proposed, which allows a Gauss-Green theorem when the vector field fails to be continuous on a compact set of (m-1)-dimensional Hausdorff measure zero and fails to be differentiable on a compact set which is the countable union of compact sets whose (m-1)-dimensional Hausdorff measures are finite. The variational integral defined in such a way has a number of interesting properties like the invariance with respect to smooth changes of coordinates, which makes it suitable for integration on manifolds. This integral combines the approaches of Kurzweil and Henstock based upon Riemann sums and Marik extension process of forming improper integrals.
scientific article; zbMATH DE number 4154781