On the continuity of the Volterra variational derivative

AbstractWe give a sufficient condition for the continuity of the Volterra variational derivative of a functional with respect to a fixed function. For linear functionals this condition is automatically satisfied, and so the Volterra variational derivative of a linear functional is always continuous

The Gauss–Green theorem and removable sets for PDEs in divergence form

AbstractApplying a very general Gauss–Green theorem established for the generalized Riemann integral, we obtain simple proofs of new results about removable sets of singularities for the Laplace and minimal surface equations. We treat simultaneously singularities with respect to differentiability and continuity

The Divergence Theorem and Sets of Finite Perimeter

This book is devoted to a detailed development of the divergence theorem. The framework is that of Lebesgue integration -- no generalized Riemann integrals of Henstock--Kurzweil variety are involved. In Part I the divergence theorem is established by a combinatorial argument involving dyadic cubes. Only elementary properties of the Lebesgue integral and Hausdorff measures are used. The resulting integration by parts is sufficiently general for many applications. As an example, it is applied to removable singularities of Cauchy--Riemann, Laplace, and minimal surface equations. The sets of fini

Special relativity without physics

Using only causality and the constant speed of light, I derive the Poincaré transformation group. In this derivation I make no a priori assumptions about the linearity or continuity of the transformation

Lectures on geometric integration and the divergence theorem

In these lectures I shall present some geometric aspects of the generalized Riemann integral, defined by Henstock and Kurzweil about thirty years ago. In particular, I shall discuss in considerable detail the develop¬ment of ideas that led to a multidimensional version of the integral, which is coordinate free and provides the divcigence theorem for nonlipschitzian vector fields. No a priori knowledge of the subject is assumed

On variations of functions of one real variable

summary:We discuss variations of functions that provide conceptually similar descriptive definitions of the Lebesgue and Denjoy-Perron integrals

When absolutely continuous implies a-finite

We show that the variational measure of an additive continuous function of figures is cr-finite whenever it is absolutely continuous. This result yields a new descriptive definition of a multidimensional generalized Riemann integral.Buczolich Zoltân, Pfeffer Washek F. When absolutely continuous implies a-finite. In: Bulletin de la Classe des sciences, tome 8, n°1-6, 1997. pp. 155-160