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 
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 
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