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The Divergence theorem and sets of f...

Pfeffer, Washek F.

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The Divergence theorem and sets of finite perimeter /

Record Type:	Language materials, printed : Monograph/item
Title/Author:	The Divergence theorem and sets of finite perimeter // Washek F. Pfeffer.
Author:	Pfeffer, Washek F.
Published:	Boca Raton :CRC Press, : c2012.,
Description:	xv, 242 p. :ill ;25 cm.
Subject:	Differential calculus. -
ISBN:	9781466507197 (hbk.) :

The Divergence theorem and sets of finite perimeter /
Pfeffer, Washek F.

The Divergence theorem and sets of finite perimeter /Washek F. Pfeffer. - Boca Raton :CRC Press,c2012. - xv, 242 p. :ill ;25 cm. - Monographs and textbooks in pure and applied mathematics.

Includes bibliographical references (p. 231-233) and index.

"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 one-dimensional 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"--

ISBN: 9781466507197 (hbk.) :UK63.99

LCCN: 2012005948Subjects--Topical Terms:

648417
Differential calculus.

LC Class. No.: QA433 / .P493 2012

Dewey Class. No.: 515/.4

The Divergence theorem and sets of finite perimeter /
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020 $a 9781466507197 (hbk.) : $c UK63.99
020 $a 1466507195 (hbk.)
020 $a 9781466507210 (ebk.- PDF)
020 $a 1466507217 (ebk.-PDF)
035 $a AS-BW-102-N-03
040 $a DLC $c DLC $d DLC
050 0 0 $a QA433 $b .P493 2012
082 0 0 $a 515/.4 $2 23
100 1 $a Pfeffer, Washek F. $3 604844
245 1 0 $a The Divergence theorem and sets of finite perimeter / $c Washek F. Pfeffer.
260 # $a Boca Raton : $b CRC Press, $c c2012.
300 $a xv, 242 p. : $b ill ; $c 25 cm.
490 0 $a Monographs and textbooks in pure and applied mathematics
504 $a Includes bibliographical references (p. 231-233) and index.
520 # $a "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 one-dimensional 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"-- $c Provided by publisher.
650 # 0 $a Differential calculus. $3 648417
650 # 0 $a Divergence theorem. $3 2012535