The divergence theorem and sets of finite perimeter
 Author/Creator
 Pfeffer, Washek F.
 Language
 English.
 Imprint
 Boca Raton : CRC Press, c2012.
 Physical description
 xv, 242 p. : ill. ; 25 cm.
 Series
 Monographs and textbooks in pure and applied mathematics ; 303.
Access
Available online

Stacks

Unknown
QA433 .P493 2012

Unknown
QA433 .P493 2012
More options
Contents/Summary
 Bibliography
 Includes bibliographical references (p. 231233) and index.
 Summary
 "Preface The divergence theorem and the resulting integration by parts formula belong to the most frequently used tools of mathematical analysis. In its elementary form, that is for smooth vector fields defined in a neighborhood of some simple geometric object such as rectangle, cylinder, ball, etc., the divergence theorem is presented in many calculus books. Its proof is obtained by a simple application of the onedimensional fundamental theorem of calculus and iterated Riemann integration. Appreciable difficulties arise when we consider a more general situation. Employing the Lebesgue integral is essential, but it is only the first step in a long struggle. We divide the problem into three parts. (1) Extending the family of vector fields for which the divergence theorem holds on simple sets. (2) Extending the the family of sets for which the divergence theorem holds for Lipschitz vector fields. (3) Proving the divergence theorem when the vector fields and sets are extended simultaneously. Of these problems, part (2) is unquestionably the most complicated. While many mathematicians contributed to it, the Italian school represented by Caccioppoli, De Giorgi, and others, obtained a complete solution by defining the sets of bounded variation (BV sets). A major contribution to part (3) is due to Federer, who proved the divergence theorem for BV sets and Lipschitz vector fields. While parts (1)(3) can be combined, treating them separately illuminates the exposition. We begin with sets that are locally simple: finite unions of dyadic cubes, called dyadic figures. Combining ideas of Henstock and McShane with a combinatorial argument of Jurkat, we establish the divergence theorem for very general vector fields defined on dyadic figures" Provided by publisher.
Subjects
Bibliographic information
 Publication date
 2012
 Responsibility
 Washek F. Pfeffer.
 Series
 Monographs and textbooks in pure and applied mathematics ; 303
 Note
 "A Chapman & Hall book."
 ISBN
 9781466507197
 1466507195