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 sort are involved.
In half I the divergence theorem is confirmed through a combinatorial argument related to dyadic cubes. merely easy houses of the Lebesgue critical and Hausdorff measures are used. The ensuing integration by means of elements is satisfactorily basic for lots of functions. as an instance, 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 received through the features 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 services of bounded variation.
The basic divergence theorem for bounded vector fields is proved partially III. The facts 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 includes a characterization of the distributions which are equivalent to the flux of a continual vector field.
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