By Washek F. Pfeffer
This e-book is dedicated to a close improvement of the divergence theorem. The framework is that of Lebesgue integration — no generalized Riemann integrals of Henstock–Kurzweil kind are involved.
In half I the divergence theorem is verified via a combinatorial argument concerning dyadic cubes. purely ordinary homes of the Lebesgue quintessential and Hausdorff measures are used. The ensuing integration by means of components is adequately common for lots of functions. to illustrate, 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 provided. The equivalence of those viewpoints is got through the capabilities of bounded version. those features are studied in a self-contained demeanour without references to Sobolev’s areas. The coarea theorem presents a hyperlink among the units of finite perimeter and capabilities of bounded variation.
The basic divergence theorem for bounded vector fields is proved partly III. The facts involves 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 continual vector field.
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