Vol. 4 No 1 - Pi Mu Epsilon

calculus gg Variations. By I. M. Gelfand and S. V. Fomin. Translated andedited by R. A. Silverman. Englewood Cliffs, N. J., Prentice-Hall, 1963.vii + 232 pp., $7.95.This is an attractive and modern treatment of the calculus ofvariations written by two eminent Soviet mathematicians. This (authorized)translation includes exercises and two appendices not in the originalRussian edition. It should be very useful for'itudents who wish to learnthis interesting and Important field and who have a background in realanalysis.The problems of the calculus of variations are very old and go backto the Bernoullis, Newton, and Euler. Important contributions were latermade by Hamilton, Jacobi, Hilbert, and Weierstrass, among others. Yetthere is still much research activity on various aspects of the theory.The main theory of the subject is well-presented. Thus, necessaryand sufficient conditions for an extreme are discussed, as are the~amilton-Jacob! theory, fields and conjugate points, and variationalproblems involving multiple integrals. Applications to mechanics andphysics are also indicated.One of the most attractive features of this book to this reviewer isthat the authors approach the subject from the point of view of modernfunctional analysis. This sets it off from most other books that areavailable on this subject.university of Illinois Robert G. BartleIntroduction to Set Theory and Logic. By Robert R. StOll. San Francisco:W. H. Freeman and Company, 1963. xiv + 474 pp., $9.00.This book was written as a text book for a year course for advancedundergraduates to give them initial training in the axiomatic methodof mathematics. The title notwithstanding, the main focus is upon thereal number system. There is ample material for a year course in "TheFoundations of Mathematics," so that the instructor may do some selecting,choosing, and emphasizing to suit his own desires. The beginning chaptersintroduce the three areas of set theory, real number development, andlogic. Further chapters go into all three areas much more extensively,and are independent enough so that the book may be used as a one semestertext for any of the three areas. The format is very good, with definedwords in bold face type and the words "Theorem", "Lemma", etc. in verylarge and distinctive type. There are numerous examples throughout, andquite a sufficient number of exercises for most sections to be in keepingwith the text book aim.m, Relations, & Functions: & Programed Unit in Modern Mathematics.By M. McFadden, J. W. Moore, and W. I. Smith. New York, McGraw-Hill BookCompany, Inc., 1963. iv + 299 pp., $3.95 paper, $5.95 cloth.This book is a programmed unit in modern mathematics geared for thosewho desire to increase their understanding of the "why" in mathematics andfor those who have finished two years of high school algebra. The matteris presented in the form of frames: one problem to a frame, with theanswer written immediately below in a shaded area. Each frame requiresthe reader to answer the problem presented either with fill-ins, drawingconstructions, or graphs. On the average there are four frames to a page.In all there are 1074 individual frames, not counting the nine self-tests.The answers to these self-tests are given at the end of the book.The title indicates the three main divisions of the book. In thesection on sets are considered the description and notation of sets,operations with sets, and equations and inequalities involving sets. Thesection on relations treats ordered pairs of numbers, graphs, cartesianproduct, and binary relations. Various mappings, value of a function,composite function, inverse functions, and various types of correspondencesconstitute the subject matter of the section on functions.Each new topic is introduced by giving many examples, and only afterthese are discussed is a formal definition given. However, some notionsare used without any formal definition at all: e.g., a one-to-one correspondencein frame 100 is undefined, although it is used in the definitionof equivalent sets; finite and infinite sets in frame 121 are not defined;the relation of "greater than" in frame 318 is left without a definition.The method of introducing ordered pairs might lead one to think that onlypositive numbers are considered for the elements of each pair, thoughlater on this impression is removed.Much emphasis is placed on the use of Venn diagrams to exemplifyproperties of sets and operations with sets. For one using the text withouta teacher, false impressions could be readily formed that Venn diagramsprove the various properties of sets. Formal proofs of some mathematicalproperties are given and occur as problems in one or more successiveframes. The review frames and the self-tests are quite appropriate.There are some misprints which would be obvious to those studyingthe book with the help of a teacher, but for one going through on his own,they may not be so obvious: e.g., frames 161, 283. Most of the practicalapplications of the theory occurs in the latter part of the book afterfunctions are treated.For one who works through the entire book a fine knowledge of sets,relations, and functions with their fundamental properties would be hisreward.DePauw University Robert J. Thomas -