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Fundamentals of Advanced Mathematics V3 (New Mathematical Methods, Systems and Applications) PDF

414 Pages·2019·9.807 MB·English
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Fundamentals of Advanced Mathematics 3 New Mathematical Methods, Systems and Applications Set coordinated by Henri Bourlès Fundamentals of Advanced Mathematics 3 Differential Calculus, Tensor Calculus, Differential Geometry, Global Analysis Henri Bourlès First published 2019 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Press Ltd Elsevier Ltd 27-37 St George’s Road The Boulevard, Langford Lane London SW19 4EU Kidlington, Oxford, OX5 1GB UK UK www.iste.co.uk www.elsevier.com Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. For information on all our publications visit our website at http://store.elsevier.com/ © ISTE Press Ltd 2019 The rights of Henri Bourlès to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress ISBN 978-1-78548-250-2 Printed and bound in the UK and US Preface This third volume of Fundamentals of Advanced Mathematics (the first two volumes are referenced by [P1] and [P2] below) is dedicated to differential and integralcalculus,examinedfromboththelocalandglobalperspectives.Thisbookis intended for anyone who uses mathematics (mathematicians, but also physicists and engineers,andinparticularanyonewhoneedstounderstandthecontrolofnonlinear systems). Some local questions of integral calculus were already partially addressed in[P2],Chapter4,andthenaturalframeworkofdifferentialcalculus,Banachspaces, ispresentedinChapter1ofthisthirdvolume.Nonetheless,wewillneedtoconsider a few generalizations to sketch the so-called “convenient” context for more recent developments in global analysis; more on this later. We will also present the “Carathédory conditions”, which are finer than the classical Cauchy–Lipschitz existence and uniqueness conditions for solutions of finite-dimensional ordinary differentialequations. Globalquestionsdemandanotherperspective.Seenthroughourwindow,theEarth appearsentirelyflat, yeteventheGreeksintheageofPlatoknewbetter, astestified byanexcerptfromhisPhaedo(108,e).KnowledgeoftheEarth’sshapeundoubtedly extended back to the Pythagoreans in the 6th Century BC. Thus, Euclid, who was an avid student of Plato’s work (consider, for example, his construction of the five Platonic solids in Book XIII of the Elements), certainly also knew that the Earth is round. Yet his geometry is quite different from the geometry of a sphere. Euclidean geometryoffersagoodlocalapproximationofsphericalgeometrybutwasofcourse useless for the lengthy voyages of the Renaissance. Global analysis must therefore beexpressedintheframeworkofmanifolds,aconceptwhichgeneralizescurvesand surfacessinceRiemann. Ever since the invention of variational calculus in the late 17th Century, it has become common in mathematics to argue about sets of functions. If certain conditions are satisfied, these sets can be endowed with the structure of a manifold, xii FundamentalsofAdvancedMathematics3 in which case they are called “functional manifolds” and imagined as deformed versions of the usual function spaces (in the same way that the Earth might be imagined as a deformed version of the plane). “Banach” manifolds were the first candidatestobeconsideredasaframeworkforglobalanalysisinthelate1950sand the 1960s [EEL66, PAL68] (these manifolds are deformed versions of a Banach space). However, as became clear in [P2], section 4.3, many of the function spaces encounteredinpracticearenotBanachspaces.Forexample,thespaceE ofinfinitely differentiablefunctionsonanon-emptyopensubsetofRnisanuclearFréchetspace. Sincethe1980s,thishasinspiredresearchintomanifoldsthataredeformedversions ofspacesofthistype; thisisthe“convenient”contextforglobalanalysismentioned above(whichmaturedaroundthelate1990s[KRI97]).Althoughwewillnotbeable to present it exhaustively in this book, our discussion of manifolds of mappings in section5.3willdemonstratetheconsiderablevalueofthisapproach. Chapters2to4developtherequiredformalism,withabriefdetourinChapter3to introduceanotionthathasplayedafundamentalroleeversinceÉ.Cartan,theconcept of fiber bundles, and in particular principal bundles. According to general relativity, we live in a pseudo-Riemannian space that is the “base” of a principal bundle; the latterisnamelythemanifoldoforthonormalframe,andits“structuralgroup”(which performschangesofreferenceframe)isa“Liegroup”,namelytheLorentz–Poincaré orthogonalgroupofmatricesleavinginvariantthequadraticform(ds)2 =c2(dt)2− (dx)2−(dy)2−(dz)2.Tensorcalculus,astapleofphysicstextbookssincetheearly 20thCentury,ispresentedinChapter4,alongsidethetheoryofdifferentialp-forms. Our formalism first begins to truly bear fruit in Chapter 5. Distributions, and the generalized notion of currents, may now be defined on manifolds instead of open subsetsofRn.Theideaofexteriorderivativesofadifferentialp-form(introducedby É.Cartan)allowsustogivehighlycondensedexpressionsfortheclassicalformulas of “vector calculus” involving gradients, divergences, Laplacians, etc. The first fundamental result of this chapter is a general formulation of Stokes’ theorem encompassingtheOstrogradsky,Gauss,Green–RiemannandGreentheoremswidely used in physics, as well as the “classical” Stokes’ theorem. On a Riemannian manifold, Stokes’ theorem enables us to formulate Hodge duality, which simplifies many of our calculations involving vectors. From the perspective of algebraic topology, Stokes’ theorem also gives rise to two other types of duality: Poincaré dualityforhomologiesandDeRhamdualityforcohomologies.OnR3,forinstance, −→ weknowthatthecurlofanyvectorfield E thatderivesfromapotentialiszeroand (cid:2)→ the divergence of any vector field B that can be expressed as a curl is also zero. Stokes’ theorem allows us to prove the converse of each claim. The second fundamentalresultofChapter5istheFrobeniustheorem,whichgivesnecessaryand sufficientconditionsfortheintegrabilityofa“contactdistribution”.Thisallowsusto Preface xiii establish the concept of foliation. The Frobenius theorem also implies a result by Riemann in Chapter 7 that is essential for general relativity, namely that a Riemannian manifold is flat if and only if its curvature tensor is zero (section 7.4.3, Theorem7.56). Lie groups are manifolds but also groups; in Chapter 6, the group structure enablesustoperformoperationsthatwouldnotmakesenseonanordinarymanifold, specifically convolution of functions or distributions. Furthermore, a “taxonomy” of Lie groups can be established from the Lie algebras associated with each group, which are vector spaces and therefore easier to study: as a set, the Lie algebra g = Lie(G) of the Lie group G is the tangent space Te(G) of G at the point e, where e is the neutral element of G. However, the three “fundamental theorems” of S.Lieimplythatthereexistsa“dictionary”thatallowsustocharacterizeLiegroups by the properties of their Lie algebras, at least locally (and globally if G is simply connected).TheclassificationestablishedbyLieiscompleteinthecaseofsimpleor semi-simpleLiegroups(oralgebras).Thisisthemostimportantcase,sincetheseare the groups frequently encountered in particle physics, where they play an essential role (including the so-called “exceptional” simple Lie algebras). Simple and semi-simple Lie algebras have been studied since Cartan in terms of their “root systems”; the ability to represent these root systems graphically (as proposed by Coxeter and Dynkin, among others) is extremely useful, but cannot be presented in detailinthisbook1. On a reductive Lie group G, we can also fully develop the theory of harmonic analysis(Fouriertransformsoffunctionsortempereddistributions).Theabeliancase will be presented in detail: when G = Rn, we recover the usual notion of Fourier transform; when G is the torus Tn, we recover the Fourier series expansion of periodic functions or distributions. The non-abelian case would fill another entire volume and thus will only be briefly mentioned (even though engineering applicationshaverecentlybeenfound[CHI01]).Readersarewelcometorefertothe bibliographyforthenon-abeliancase[VAR77,VAR89]. Definingageometryonamanifoldisequivalenttoequippingthismanifoldwith a connection (Chapter 7). Lie groups are implicitly equipped with a connection. Riemannianorpseudo-Riemannianmanifoldsareoftenimplicitlyequippedwiththe simplest possible connection: the Levi-Cività connection. This is a special case of a “G-structure” that is frequently used in general relativity. É. Cartan clarified the notion of connection; he studied affine, projective and conformal connections, summarizing his ideas by proposing the concept of “generalized space” [CAR26]; 1SeetheWikipediaarticleonCoxeter–Dynkindiagrams. xiv FundamentalsofAdvancedMathematics3 these spaces are equipped with connections called Cartan connections since Ehresmann (who rephrased these ideas within the context of principal connections). Connections can be equipped with curvature (an idea that should be familiar to relativistic physicists) and in some cases torsion, which attracted considerable interestfromEinstein([EIN54],AppendixII),whohopedtofindawaytounifythe theoriesofgravitationandelectromagnetism. HenriBOURLÈS June2019 Errata for Volume 1 and Volume 2 Volume1(Cont’d) 1)Onp.12,thefifthlineof(V)shouldreadRinsteadofR. 2)Onp.22,ontheright-handsideof[1.6],itshouldreadl←im−insteadofl−i→m. 3)On p. 41, line 10 should read “the cardinal of G/H (equal to the cardinal of G\H)”insteadof“thiscardinal”. 4)Onp.45,thefirstlineafter[2.12]shouldreadM3insteadofG. 5)Onp.190,inDefinition3.177,itshouldreadΔninsteadofΔn. 6)Onp.191,inthefirstlineafter[3.70],add“alsodenotedasdp”after“operator”. 7)Onp.193,line17shouldreadSpinsteadofSn. (cid:2) 8)Onp.220,line7shouldread“π = j∈Jπj wheretheelementarydivisorsπj arepairwisenon-associatedandmaximal(cid:2)powers(amongallelementarydivisors)of irreduciblepolynomials”insteadof“π = ni=1πi”. Volume2 1)Onp.12,line6shouldread“K(cid:3)(α(cid:3))”insteadof“K(α(cid:3))”. 2)Onp.17,line19shouldread“(cid:4)=”insteadof“=”;line21shouldread“doesnot exist”insteadof“exists”. 3)Onp.20,line11shouldread“0≤i≤r”insteadof“1≤i≤r”. 4)Onp.24,line28shouldread“x∈K¯”insteadof“x∈K”. 5)Onp.27,line10shouldread“K¯”insteadof“K”. 6)Onp.32,line3shouldread“aisnotoftheformbn,b ∈ K”insteadof“an ∈/ K”;line5shouldread“ζ”insteadof“ς”. xvi FundamentalsofAdvancedMathematics3 7)Onp.43,inline10,after“yields”,add“withj =0,...,n−2”;thelastsumof (j) (j+1) line11shouldread“u ”insteadof“u ”. i i 8)On p. 51, replace the sentence beginning line 23 by: “The coefficients c and d ar(cid:3)e free par(cid:4)ameters, the former in C, the second in C×; indeed, putting M = C t,e−t2/2 we have that GalD(N,M) = C, GalD(M,K) = C×, which yieldstheshortexactsequenceofAbeliangroups0−→C−→G−→C× −→0,in otherwordsGisanextensionofC×byC([P1],section2.2.2(II)).”. 9)Onp.57,line2shouldread“nonemptyset”insteadof“set”. (cid:5)10(cid:6))Onp.62, lines2(cid:5)7, 2(cid:6)9; onp.63, lines3, 8; onp.64, lines7, 9: itshouldread “ x(cid:3) ”insteadof“ x(cid:3) ”. j j∈J j i∈J 11)On p. 83, line 22 should read “smallest” instead of “largest” and “=” instead of“(cid:4)=”. 12)Onp.91,line28shouldread“j (cid:7)i0”insteadof“j ≥i0”. 13)Onp.92,inline19,addafterthelastsentence:“Thisextensionisthenunique.”. 14)Onp.95,line8shouldread“∀(x(cid:3),x(cid:3)(cid:3))”insteadof“∀(x(cid:3);x(cid:3)(cid:3))”. 15)Onp.105,line25shouldread“∀i(cid:7)i(cid:3)0(cid:3)”insteadof“∀i(cid:7)i0”. 16)Onp.119,line17shouldread“K”insteadof“K”. 17)Onp.128, lines1, 2shouldread“(iv)”insteadofthesecond“(iii)”and“(v)” insteadof“(iv)”. 18)Onp.132,line20shouldread“ξx0 (cid:10)→ξ”insteadof“ξ (cid:10)→ξx0”. 19)Onp.133,inline14,delete“inLcsh”. 20)On p. 135, lines 19–21 should read “re(cid:2)duced if for all i ∈ I, the projection pri(E),wherepriisthecanonicalprojection iEi (cid:2)Ei,isdenseinEi”insteadof “decreasing[...]Tj)”;inlines27–31,replaceStatement2)ofRemark3.33by:“Every projectivelimitcanbeputintothef(cid:2)ormofareducedprojectivelim(cid:7)it: ifFi =(cid:3)pri(cid:4)(E) a(cid:2)nd ψ˜ij is the restriction of ψij to iFi, then E is the subspace i(cid:4)jker ψ˜ij of iFi.”. (cid:8)(cid:9) (cid:10)21)On p. 137, the last line should read “p ni=1(cid:11)xi(cid:11)p” instead of (cid:9) “p ni=1(cid:11)xi(cid:11)P”. 22)Onp.141,inlines24,26,delete“decreasing”. 23)Onp.142,inlines10,11,delete“issurjective(ibid.)and”. 24)Onp.144,line11shouldread“boundedinF”insteadof“boundedinM”.

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