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4.1
100 reviewsTheorem of Calculus) • Measure theory • Topology • Distribution theory
Mathematics Subject Classification: 26A16 Lipschitz (Hölder) classes • 26A46 Absolutely continuous real functions in one variable • 26B20 Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.) • 28A25 Integration with respect to measures and other set functions • 28A75 Length, area, volume, other geometric measure theory • 28A78 Hausdorff and packing measures • 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) • 30G35 Functions of hypercomplex variables and generalized variables • 31B10 Integral representations, integral operators, integral equations methods in higher dimensions • 31C12 Potential theory on Riemannian manifolds and other spaces • 42B25 Maximal functions, Littlewood-Paley theory • 42B37 Harmonic analysis and PDEs • Geometric measure and integration theory, integral and normal currents in optimization • Local Riemannian geometry • Integral geometry • Differential forms in global analysis • Integration on manifolds; measures on manifolds
This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations.
Volume I establishes a sharp version of the Divergence Theorem (aka Fundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundary trace is only assumed to exist in a nontangential pointwise sense.
