Differential forms : theory and practice PDF
Preview Differential forms : theory and practice
Differential Forms Theory and Practice Differential Forms Theory and Practice Second edition by Steven H. Weintraub Lehigh University Bethlehem, PA, USA AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK OXFORD • PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 225 Wyman Street, Waltham, MA 02451, USA 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA Second edition 2014 Copyright © 2014 Elsevier Inc. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone ( 44) (0) 1865 843830; fax + ( 44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your + request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material. Notice No responsibility is assumed by the publisher 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. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. Library of Congress Cataloging-in-Publication Data Weintraub, Steven H., author. Differential forms: theory and practice / by Steven H. Weintraub. -- [2] edition. pages cm Includes index. ISBN 978-0-12-394403-0 (alk. paper) 1. Differential forms. I. Title. QA381.W45 2014 515’.37--dc23 2013035820 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library For information on all Academic Press publications visit our web site at store.elsevier.com Printed and bound in USA 14 15 16 17 10 9 8 7 6 5 4 3 2 1 ISBN: 978-0-12-394403-0 To my mother and the memory of my father Preface Differential forms are a powerful computational and theoretical tool. They play a central role in mathematics, in such areas as analysis on manifolds and differential geometry, and in physics as well, in such areas as electromagnetism and general relativity. In this book, we present a concrete and careful introduction to differential forms, attheupper-undergraduateorbeginninggraduatelevel,designedwith the needs of both mathematicians and physicists (and other users of the theory) in mind. On the one hand, our treatment is concrete. By that we mean that we present quite a bit of material on how to do computations with differential forms, so that the reader may effectively use them. On the other hand, our treatment is careful. By that we mean that we present precise definitions and rigorous proofs of (almost) all of the results in this book. We begin at the beginning, defining differential forms and show- ing how to manipulate them. First we show how to do algebra with them, and then we show how to find the exterior derivative dϕ of a differential form ϕ. We explain what differential forms really are: Roughly speaking, a k-form is a particular kind of function on k-tuples of tangent vectors. (Of course, in order to make sense of this we must first make sense of tangent vectors.) We carry on to our main goal, the Generalized Stokes’s Theorem, one of the central theorems of mathematics. This theorem states: Theorem (Generalized Stokes’s Theorem (GST)). Let M be an orientedsmoothk-manifoldwithboundary∂M (possiblyempty)and let ∂M be given the induced orientation. Let ϕ be a (k −1)-form on M with compact support. Then (cid:2) (cid:2) dϕ = ϕ. M ∂M x Preface This goal determines our path. We must develop the notion of an orientedsmoothmanifoldandshowhowtointegratedifferentialforms onthese.Oncewehavedoneso,wecanstateandprovethistheorem. The theory of differential forms was first developed in the early twentieth century by Elie Cartan, and this theory naturally led to de Rham cohomology, which we consider in our last chapter. One thing we call the reader’s attention to here is the theme of “naturality” that pervades the book. That is, everything commutes withpull-backs–thiscrypticstatementwillbecomeclearuponreading thebook—andthisenablesustodoallourcalculationsonsubsetsof Rn, which is the only place we really know how to do calculus. This book is an outgrowth of the author’s earlier book Differential Forms:AComplementtoVectorCalculus.Inthatbookweintroduced differentialformsatalowerlevel,thatofthirdsemestercalculus.The point there was to show how the theory of differential forms unified and clarified the material in multivariable calculus: the gradient of a function, and the curl and divergence of a vector field (in R3) are all “really” special cases of the exterior derivative of a differential form, and the classical theorems of Green, Stokes, and Gauss are all “really” special cases of the GST. By “really” we mean that we must firstrecasttheseresultsintermsofdifferentialforms,andthisisdone by what we call the “Fundamental Correspondence.” However, in the (many) years since that book appeared, we have received a steady stream of emails from students and teachers who used this book, but almost invariably at a higher level. We have thus decided to rewrite it at a higher level, in order to address the needs of the actual readers of the book. Our previous book had minimal prerequisites,butforthisbookthereaderwillhavetobefamiliarwith thebasicsofpoint-settopology,andtohavehadagoodundergraduate course in linear algebra. We use additional linear algebra material, oftennotcoveredinsuchacourse,andwedevelopitwhenweneedit. We would like to take this opportunity to correct two historical errorswemadeinourearlierbook.Oneofthemotivationsfordevelop- ing vector calculus was, as we wrote, Maxwell’s equations in electromagnetism. We wrote that Maxwell would have recognized vector calculus. In fact, the (common) expression of those equations in vector calculus terms was not due to him, but rather to Heaviside. Preface xi Butit isindeed the case that thisisanineteenth century formulation, and there is an illuminating reformulation of Maxwell’s equations in terms of differential forms (which we urge the interested reader to investigate).Also,Poincaré’sworkincelestialmechanicswasanother important precursor of the theory of differential forms, and in partic- ular he proved a result now known as Poincaré’s Lemma. However, thereisconsiderabledisagreementamongmodernauthorsastowhat this lemma is (some say it is a given statement, others its converse). In our earlier book we wrote that the statement in one direction was Poincaré’s Lemma, but we believe we got it backwards then (and correct now). See Remark 1.4.2. We conclude with some remarks about notation and language. Results in this book have three-level numbering, so that, for exam- ple, Theorem 1.2.7 is the 7th numbered item in Chapter 1, Section 2. The ends of proofs are marked by the symbol (cid:2). The statements of theorems,corollaries,etc.,areinitalics,soareclearlydelineated.But the statements of definitions, remarks, etc., are in ordinary type, so there is nothing to delineate them. We thus mark their ends by the symbol♦.Weuse A ⊆ B tomeanthat Aisasubsetof B,and A ⊂ B to mean that A is a proper subset of B. We use the term “manifold” to mean precisely that, i.e., a manifold without boundary. The term “manifoldwithboundary”isageneralizationoftheterm“manifold,” i.e.,itincludesthecasewhentheboundaryisempty,inwhichcaseit is simply a manifold. Steven H. Weintraub Bethlehem, PA, USA May, 2013 1 Rn Differential Forms in , I InthischapterweintroducedifferentialformsinRn asformalobjects and show how to do algebra with them. We then introduce the oper- ation of exterior differentiation and discuss closed and exact forms. Our discussion here is completely general, but for the sake of clarity andsimplicity,wewillbedrawingmostofourexamplesfromR1,R2, or R3. The special case n=3 has some particularly interesting features mathematically, and corresponds to the world we live in physically. In Section 1.3 of this chapter, we show the correspondence between differential forms and functions/vector fields in R3. 1.0 Euclidean spaces, tangent spaces, and tangent vector fields We begin by establishing some notation and by making some subtle but important distinctions that are often ignored. We let R denote the set of real numbers. Definition 1.0.1. For a positive integer n, Rn is Rn = {(x ,...,x ) | x ∈ R}. (cid:3) 1 n i In dealing with R1, R2, andR 3, we will often use (x), (x,y), and (x,y,z) rather than (x ), (x ,x ), and (x ,x ,x ), respectively, as 1 1 2 1 2 3 coordinates. We are about to introduce tangent spaces. For some purposes, it is convenient to have a single vector space that serves as a “model” for each tangent space, and we introduce that first. DifferentialForms,SecondEdition.http://dx.doi.org/10.1016/B978-0-12-394403-0.00001-3 ©2014ElsevierInc.Allrightsreserved. 2 DifferentialForms Definition 1.0.2. For a positive integer n, Rn is the vector space ⎧ ⎡ ⎤ ⎫ ⎨ a ⎬ 1 . v = ⎣ .. ⎦ | a ∈ R . ⎩ i ⎭ a n with the operations ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ a b a +b a ca 1 1 1 1 1 1 . . . . . ⎣ .. ⎦+⎣ .. ⎦ = ⎣ .. ⎦ and c⎣ .. ⎦ = ⎣ .. ⎦. (cid:3) a b a +b a ca n n n n n n Definition 1.0.3. Let p=(x ,...,x )∈Rn. The tangent space 1 n Rn = T Rn to Rn at p is p p ⎧ ⎫ ⎡ ⎤ ⎪ ⎪ ⎨ a ⎬ 1 . v = ⎣ .. ⎦ | a ∈ R . (cid:3) ⎪ p i ⎪ ⎩ ⎭ a n p Let us carefully discuss the distinction here. Elements of Rn are points, while for each fixed point p of Rn, elements v of T Rn are p p vectors. For a fixed point p of Rn, ifv ∈ T Rn, we say that v is a p p p tangentvectorbasedatthatpoint,sothat T Rn isthevectorspaceof p tangent vectors to Rn at p. This is indeed a vector space as we have the operations of vector addition and scalar multiplication given by ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ a b a +b a ca 1 1 1 1 1 1 . . . . . ⎣ .. ⎦ +⎣ .. ⎦ = ⎣ .. ⎦ and c⎣ .. ⎦ = ⎣ .. ⎦ . a b a +b a ca n p n p n n p n p n p WesaythatRn servesasamodelforeverytangentspaceRn aswe p have an isomorphism from Rn to Rn given by p ⎡ ⎤ ⎡ ⎤ a a 1 1 . . ⎣ .. ⎦ (cid:4)→ ⎣ .. ⎦ . a a n n p Geometrically, it makes sense to add tangent vectors based at the samepointofRn,andthatiswhatwehavedone.Butitdoesnotmake DifferentialFormsinRn,I 3 geometricsensetoaddtangentvectorsbasedatdifferentpointsofRn, nor does it make geometric sense to add points of Rn, so we do not attempt to define such operations. Let us emphasize the distinctions we have made. Often one sees points and vectors identified, but they are really different kinds of objects. Also, often one sees tangent vectors at different points iden- tified but they are really different objects. What we have just said is mathematically precise, but also makes sensephysically.Consider,forexample,abodyunderthegravitational influence of a star. The body has some position, i.e., it is located at some point p of R3, and the gravity of the star exerts a force on the body,thisforcebeinggivenbyavectorv whosedirectionistoward p the star and whose magnitude is given by Newton’s law of universal gravitation. Furthermore v is indeed based at p as this is where the p body is located, i.e., this is the point at which the force is acting. Thusweseeacleardistinctionbetweentheposition(point)andforce (vectorbasedatthatpoint).Ifourbody,locatedatthepoint p,isunder the influence of a binary star system, with v the force vector for the p gravitational attraction of the first star and w the force vector for p thegravitationalattractionofthesecondstar,thentheirsumv +w p p (given by the “parallelogram law”) is the net gravitational force on the body, and this indeed makes sense. On the other hand, if v is a p tangent vector based at p and w is a tangent vector based at some q different point q, their sumv + w is not defined, and would not p q make physical sense either, as what could this represent? (A body can’t be in two different places at the same time!) Nextwecometothecloselyrelatednotionofatangentvectorfield. Definition 1.0.4. A tangent vector field v on Rn is a function v that associates to every point p ∈ Rn a tangent vector to Rn based at p, i.e., a tangent vector v ∈ T Rn. Thus we may write v(p) = v p p p for every p ∈ Rn. (cid:3) Example 1.0.5. Foranyn-tupleofrealnumbers(a ,...,a ),we 1 n have the constant vector field ⎡ ⎤ a 1 . v = ⎣ .. ⎦ a n
The list of books you might like
