Review Of: Vector Calculus By Michael Corral
University of Puget SoundSound IdeasAll Faculty ScholarshipFaculty Scholarship9-1-2009Review of: Vector Calculus by Michael CorralRobert A. BeezerUniversity of Puget Sound, beezer@pugetsound.eduFollow this and additional works at: http://soundideas.pugetsound.edu/faculty pubsCitationBeezer, Robert A. 2009. "Vector Calculus." Siam Review 51(3): 642-644.This Review is brought to you for free and open access by the Faculty Scholarship at Sound Ideas. It has been accepted for inclusion in All FacultyScholarship by an authorized administrator of Sound Ideas. For more information, please contact soundideas@pugetsound.edu.
642BOOK REVIEWSof the great wealth of work on geometrical applications of minimal models that hasbeen done over the last thirty years or more.The book begins with a summary of basic results concerning Lie groups. Lie algebras, and homogeneous spaces. The secondchapter then gives a thorough introductionto Sullivan's de Rham-type complexes andtheir minimal models. The applications begin with the third chapter, which discussesthe rational homotopy theory of manifoldsand includes such interesting results as thenecessary and sufficient conditions for agiven minimal commutative graded differential algebra to be the minimal model of asimply connected closed manifold. A chapter is then devoted to complex and symplectic manifolds. Thefifthchapter studiesclosed geodesies. These are of great intrinsic interest not only in differential geometrybut also in mechanics, where they correspond to periodic solutions. Minimal models are extremely effective tools here: theyhave led to a very elegant criterion for theexistence of infinitely many geometricallydistinct closed geodesies in a Riemannianmanifold and much else besides. A lot ofthe results in this chapter concern the freeloop space of a manifold (which is also ofinterest in string theory): it is the growth ofits Betti numbers that bears on the numberof closed geodesies.The next chapter gives applications tocurvature; and the seventh chapter discusses transformation groups where minimal models have also been very useful. Theeighth chapter concerns a variety of topics including complements of submanifolds,symplectic blow-ups, and the Chas-SuUivanloop product on the homology of the freeloop space. The final chapter is a compendium of assorted further applicationstogether with a list of some major unsolvedproblems.The book is somewhat impressionistic inits style—almost a guide to the hterature—rather like Berger's enormously interestingand informative overview of Riemannian geometry [B]. It will be very useful to anyoneinterested in any of the wide range of topicscovered. It would also be a good text fora graduate seminar: there would be plentyof work for students in reading some of thequoted literature (there is an excellent listof references) andfillingin some of the details of various topics. There are exercisestoo.REFERENCES[B][BG]M. BERGER, A PanoramicView ofRiemannian Ceometry, SpringerVerlag, Berlin, Heidelberg, 2003.A. K. BousFiELD AND V. K. A. M. GuGENHEIM, On P.L. de Rham Theory and Rational Homotopy Type,Mem. Amer. Math. Soc. 179, AMS,Providence, RI, 1976.[FHT] Y. FÉLIX, S. HALPERIN, AND J . - G .THOMAS, Rational Homotopy The-ory, Grad. Texts in Math. 205,Springer-Verlag, New York, 2001.[GM]P. A. GRIFFITHS AND J. W. MORGAN,Rational Homotopy Theory andDifferential Porms, Progr. Math.16, Birkhäuser, Boston, 1981.[H][Q][S]S. HALPERIN,Lectures on MinimalModels, Mém. Soc. Math. France(N.S.) 230, Soc. Math. France,Paris, 1983.D. QuiLLEN, Rational homotopy theory, Ann. of Math. (2), 90 (1969),pp. 205-295.D. SULLIVAN, Infinitesimal computa-tions in topology, Inst. HautesÉtudes Sei. Publ. Math., 47 (1977),pp. 269-331.[W]H. WHITNEY,Geometric IntegrationTheory, Princeton University Press,Princeton, NJ, 1957.CHRISTOPHER ALLDAYUniversity of Hawaii at ManoaVector Calculus. By Michaei Corral, www.mecmath.net, 2008. 1 1.00. vi-f2l3 pp., softcover (free electronic download available).Vector Calculus is a traditional treatmentof the topics covered in a third-semestercalculus course, often called "Multivariate Calculus" or just "Cale III." Whilethe treatment may be traditional, thedistribution and publication terms are aprime example of the newest trends intextbook publication. The author has astandard copyright interest, but has usedthis government-sanctioned monopoly togrant you additional freedoms in the use
BOOK REVIEWSof his work through the choice of a GNUFree Documentation License (GFDL). Youmay download the PDF version from theauthor's website at no cost, and you mayin turn distribute it from your own site.You may make as many printed copies ofthe book as you like, and you may do thisforever. You may modify the content (viathe downloadable source files) for your personal use. The only condition imposed bythe author is that if you make modificationsand distribute the modified versions, thenyou must apply the same license. By thisarrangement, any improvements are madeavailable for the common good.You may, of course, carefully scrutinizethis text in as much detail as you desire bydownloading the PDF version. Suffice it tosay that the topics chosen are very standardfor this course. The first chapter establishesthe basics of vector algebra, lines and planesin M , and basic calculus of vector-valuedfunctions. The second chapter is centeredon partial derivatives of functions of twoor more variables, culminating with unconstrained and constrained optimization. Thenext chapter begins with a concise treatment of double and triple integrals, includesa welcome section on numerical techniquesfor multiple integrals (Java source code fora Monte Carlo method is included), and finishes with applications to center of mass andprobability distributions. The final chapterconsiders line and surface integrals with theusual discussions of Green's theorem andStokes' theorem. The only expected topicI found not fully represented is curvature,which appears in the exercises for the firstchapter.The writing style is crisp and concise,while being both informal in tone and accurate in the technical details. Theorems arestated carefully and set ofF from surrounding text, often with proofs, subtleties arediscussed, and pathologies are mentionedbut not belabored. In the preface, the author states that on a 1 to 10 scale, witha 1 being "completely informal" and a 10being "completely rigorous," he rates thebook as a 5. When finer points do arise,a footnote will point to one of the moreadvanced texts described in the excellentannotated bibliography. A refreshing benefit of open-content books is authors striving643for the intellectual honesty they feel a topicdeserves, without being driven by marketing concerns. This book is a good exampleof striking the right balance. I would expect a student in a course to find the textvery readable in conjunction with lectures,and the independent student should findthe explanations sufficient to gain an understanding without assistance.Many of the crop of new open-contenttexts are noticeably deficient in exercises,figures, and the use of color. This texthas 420 exercises, about 15 per section,graded as "Easy," "Moderate," and "Challenging." As an instructor, after pickingand choosing the problems that suit theaims of my course, I think I would findthe need to augment the provided exerciseswith a few of my own, especially to provide some more challenging exercises (thereseems to be very good coverage of routineand drill-type problems). Of course, withthe source files available, it is nearly trivialto incorporate your own additions to theproblem sets. For a student's first exposureto functions of several variables, visualizations of surfaces, tangent planes, vectors,volumes, etc., are crucial, and this text doesnot disappoint. The excellent graphics arebuilt and rendered with the standard tools:MetaPost, PGF, and Cnuplot. Full color isused effectively in the PDF version, whilestill printing accurately in grayscale, bothin the graphics and in the highlighting ofdefinitions, theorems, and corollaries. Anappendix includes a careful tutorial for thestudent on the use of the open-source plotting package Gnuplot. Other open-contentauthors could learn something by carefullystudying Corral's example.It is important to note that this text isnot an e-book, not an online book, and nota distance-learning resource. It is conceivedand designed to be a book. But its promotion and distribution are made possible bycomputer networks, without an investmentof capital in a print run. No need to orderan evaluation copy, just download the entiretext. Only need a single chapter, a singlesection, or a single application? Just printthe subset you need (while acknowledgingthe author and the license terms). Shouldyou desire a printed copy, and for a course Ithink a student would want a printed copy.
644BOOK REVIEWSit may be ordered via the print-on-demandservice at lulu.com. The cost for a softcover, grayscale version is merely 11.00(plus shipping), which represents the entireproduction cost at Lulu, with no royaltiesgoing to the author. The sourcefilesfor thebook (I T X and MetaPost files) may beeasily downloaded. It would appear that acouple of additional packages must be addedto a standard 1 distribution in order tosuccessfully compile the book through thesupplied shell script, but this is a smallinconvenience to achieve the high-qualityoutput and graphics. The source appearsto be written in a style that would makemodifications easy for someone reasonablyfamiliar with T K.It is refreshing to see science and mathematics return to a free exchange of ideas,without proprietary interests impeding arapid and easy interchange. Corral shouldbe applauded for his excellent and unselfishcontribution to this movement. An instructor who builds a course around this text willnever need to confront the unfortunate decision to take it out of print or the release ofa new edition modified simply to frustratethe used-book market. Based on contentalone, this text is worth consideration fora traditional multivariate calculus course,is perfect for a quick refresher, is a goodchoice as a supplemental text for a moreadvanced course (such as probability), andis a great choice for the independent student on a budget. Additionally, the price isright and the extra inherent freedoms are awelcome novelty.ROBERT A .BEEZERUniversity ofPugetSoundLegacy of the Luoshu: The 4,000 YearSearch for the Meaning of the MagicSquare of Order Three. By Frank J. Swetz.A K Peters, Wellesley, MA, 2008. 39.00.xiv-h2l4 pp., hardcover. ISBN 978-1-56881427-8.Mathematics is frequently used to understand the world around us. It has beenconcerned not just with logical truth, butalso with beauty and meaning. In thisbook Swetz shows us that despite culturaldifferences and advances in knowledge, forthousands of years its role has been centralin understanding more of the world than weperceive directly. Swetz traces the historyof one simple mathematical structure, themagic square of order three, across centuriesand continents. As promised he reveals acultural history "that touches on cosmology, mythology, philosophy, religion, occultpractices, mathematics, architecture, andeven music."438951276The magic square of order three, or Luoshu, is a simple but elegant structure.Adding elements in any row, any column, orthe front or back diagonal, one obtains thesame sum, 15. Both amateur and professional mathematicians have generalized this"magic" property to larger sizes, more dimensions, and to circles and spirals insteadof squares and cubes. Swetz introduces thereader to these many generalizations andprovides a starting point for a study of themathematics of magic squares. But this isnot his main focus.Instead, he takes the reader back to thehistorical origins of the Luoshu and revealsmany interpretations of its meaning. He introduces us to widely held beliefs that thenumbers 5 and 9 have a mystical significance. While admitting that many modernreaders find such beliefs to be "claptrap,"he succeeds admirably in helping the readerto understand them. Indeed the reader appreciates how such a simple object couldbe seen to be truly magical, a window intounderstanding the world and the heavensbeyond. By developing the context in whichthe educated people of ancient China interpreted the Luoshu, Swetz lays the foundation for its impact on the practices of theimperial court in China. He traces its evolution within the yinyang theory, with fengshui, and with Daoism, among others. Itsuse in forecasting the future and ensuringfuture success may seem implausible today,but the reader cannot fail to see that theseideas had a remarkable coherence and relevance to the people of the time.Swetz also leads the reader on a journey following the Luoshu through Southeast
Vector Calculus. By Michaei Corral, www. mecmath.net, 2008. 1 1.00. vi-f2l3 pp., soft-cover (free electronic download available). Vector Calculus is a traditional treatment of the topics covered in a third-semester calculus course, often called "Multivari-ate Calculus" or just "Cale III." While the treatment may be traditional, the
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