MARC 主機 00000cam 22004934a 4500
001 758394860
003 OCoLC
005 20120927224038.0
008 120312s2012 flu b 001 0 eng
010 2012005948
020 9781466507197 (hardback)
020 1466507195 (hardback)
035 (OCoLC)758394860
040 DLC|beng|cDLC|dYDX|dBTCTA|dYDXCP|dBWX|dOCLCO|dUKMGB|dIG#
|dCOO|dVRC|dCDX|dAS|dMATH
042 pcc
050 00 QA433|b.P493 2012
082 00 515/.4|223
090 QA433/P493/2012/////52446
100 1 Pfeffer, Washek F
245 14 The divergence theorem and sets of finite perimeter /
|cWashek F. Pfeffer
260 Boca Raton :|bCRC Press,|c2012
300 xv, 242 p. ;|c25 cm
490 1 Monographs and textbooks in pure and applied mathematics
504 Includes bibliographical references (p. 231-233) and index
520 "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"--|cProvided by publisher
650 0 Divergence theorem
650 0 Differential calculus
650 7 MATHEMATICS / Advanced.|2bisacsh
650 7 MATHEMATICS / Differential Equations.|2bisacsh
650 7 MATHEMATICS / Functional Analysis.|2bisacsh
830 0 Monographs and textbooks in pure and applied mathematics