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```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
Combining ideas of Henstock and McShane with a
combinatorial argument of Jurkat, we establish the
divergence theorem for very general vector fields defined
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