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1、Chapter6DifferentiationInthischapterwelookattwoaspectsoftherelationshipbetweendifferentiationandintegration.First,inSect.6.1,welookatchangesofvariablesind-dimensionalintegrals.Suchchangesofvariablesoccur,forexample,whenoneevaluatesanintegraloveraregioninR2byconvertingtopolarcoordinates.
2、Then,inSects.6.2and6.3,welookatsomedeeperaspectsofdifferentiationtheory,includingthealmosteverywheredifferentiabilityofmonotonefunctionsandofindefiniteinte-gralsandtherelationshipbetweenRadonNikodymderivativesanddifferentiationtheory.TheVitalicoveringtheoremisanimportanttoolforthis.Thedi
3、scussionofdifferentiationtheorywillberesumedwhenwediscusstheHenstockKurzweilintegralinAppendixH.6.1ChangeofVariableinRdInthissectionwedealwithchangesofvariableinRdandwiththeirrelationtoLebesguemeasure.ThemainresultisTheorem6.1.7.Letusbeginbyrecallingsomedefinitions.LetMdbethesetofalldbyd
4、matriceswithrealentries,andletDbeareal-valuedfunctiononMd.WewillsometimesfinditconvenienttodenotethecolumnsofadbydmatrixAbyA1,A2,...,AdandtowriteD(A1,A2,...,Ad)inplaceofD(A).ThefunctionDismultilinearifforeachiandeachchoiceofAj(forj=i)themapAi→D(A1,...,Ad)islinear,isalternatingifD(A)=0h
5、oldswhenevertwoofthecolumnsofAareequal,andisadeterminantifitismultilinear,isalternating,andsatisfiesD(I)=1(hereIis,ofcourse,thedbydidentitymatrix).Weneedtorecallafewbasicfactsaboutdeterminants.Lemma6.1.1.ForeachpositiveintegerdthereisauniquedeterminantonMd.Wefollowthestandardusageandused
6、et(A)todenotethedeterminantofamatrixA.D.L.Cohn,MeasureTheory:SecondEdition,BirkhauserAdvanced¨155TextsBaslerLehrb¨ucher,DOI10.1007/978-1-4614-6956-86,©SpringerScience+BusinessMedia,LLC20131566DifferentiationLemma6.1.2.Letdbeapositiveinteger,andletMdbethesetofalldbydmatriceswithrealentri
7、es.Then(a)det(AB)=det(A)det(B)holdsforallA,BinMd,(b)det(A)isnonzeroifandonlyifAisinvertible,(c)det(A)isapolynomialinthecomponentsofA,and(d)det(At)=det(A),whereAtisthetransposeofA.ProofsofLemmas6.1.1and6.1.2canbefoundinHalmos[53]andHoffmanandKunze[61].RecallthatifT:Rd→Rdislinear
