By Washek F. Pfeffer
This booklet is dedicated to an in depth improvement of the divergence theorem. The framework is that of Lebesgue integration — no generalized Riemann integrals of Henstock–Kurzweil type are involved.
In half I the divergence theorem is validated through a combinatorial argument regarding dyadic cubes. in simple terms straightforward houses of the Lebesgue fundamental and Hausdorff measures are used. The ensuing integration through elements is satisfactorily common for lots of functions. for example, it really is utilized to detachable singularities of Cauchy–Riemann, Laplace, and minimum floor equations.
The units of finite perimeter are brought partly II. either the geometric and analytic issues of view are awarded. The equivalence of those viewpoints is bought through the services of bounded edition. those features are studied in a self-contained demeanour with out references to Sobolev’s areas. The coarea theorem offers a hyperlink among the units of finite perimeter and features of bounded variation.
The common divergence theorem for bounded vector fields is proved partly III. The evidence comprises adapting the combinatorial argument of half I to units of finite perimeter. The unbounded vector fields and suggest divergence also are mentioned. the ultimate bankruptcy incorporates a characterization of the distributions which are equivalent to the flux of a continuing vector field.
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The Divergence Theorem and Sets of Finite Perimeter (Chapman & Hall/CRC Pure and Applied Mathematics) by Washek F. Pfeffer