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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Differential equations
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The Divergence Theorem and Sets of Finite Perimeter (Paperback)
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The Divergence Theorem and Sets of Finite Perimeter (Paperback)
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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 finite perimeter are introduced in Part II.
Both the geometric and analytic points of view are presented. The
equivalence of these viewpoints is obtained via the functions of
bounded variation. These functions are studied in a self-contained
manner with no references to Sobolev's spaces. The coarea theorem
provides a link between the sets of finite perimeter and functions
of bounded variation. The general divergence theorem for bounded
vector fields is proved in Part III. The proof consists of adapting
the combinatorial argument of Part I to sets of finite perimeter.
The unbounded vector fields and mean divergence are also discussed.
The final chapter contains a characterization of the distributions
that are equal to the flux of a continuous vector field.
General
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