BOOK REVIEWS 351

in a very nice way; for example, to discover where Hall’s theorem is discussed,through the bibliography one can find the page of the text on which it is discussed.

Even the exercises are pointers to reference material and as such are helpful.However, the reader should be warned before undertaking an enterprise such assolving those exercises at the end of a given chapter. A few actually deserve thename, but most are difficult, being the main object of the referenced researchpapers.

In general the book has a lightness of style that is attractive. Like everythingtouched by Harary, his personality comes through on every page. When the basicterminology is introduced ("Some authors actually define a ’graph’ as a graph"),or when the four color problem is discussed ("The Four Color Conjecture cantruly be named the ’Four Color Disease’ for it exhibits so many properties of aninfection."), one cannot help but chuckle. Indeed, humor is a rare quality inmathematical writing. One can learn ofthe origins ofgraph theory and the "inside"knowledge, e.g., the identity of the mysterious lady Blanche Descartes. And onecan learn the mysteries of "Michigan notation," that terminology used by theauthor and his many disciples.

The book does not, and does not pretend to, include complete treatment ofspecialized topics, such as matroids, directed graphs, planar graphs, connectivityor enumeration. These are left for other books. Broad coverage, bright yetauthoritative, is its attribute.

D. H. YOUNGERUniversity of Waterloo

Mehrdimensionale Variationsrechnung.* By R. KLOTZLER. Birkhauser Verlag,Basel, Switzerland, 1970.249 pp. 54 Swiss francs.

The present volume is the first book on the calculus ofvariations to be devotedmainly to multiple integral problems of Bolza type. These are multiple integralproblems involving differential constraints and are among the most difficultproblems in the calculus of variations. The author has made notable contributionstowards the development of a theory comparable in completeness with that forsimple integral problems of Bolza. In particular the theory with regard to weakrelative minima is fully developed. Except for the quadratic case there remains theusual gap between the necessary conditions and sufficient conditions for a strongrelative minimum.

The present book is not concerned with general existence theorems. Theauthor restricts himself to the study of necessary conditions for a minimum, fieldtheory and sufficient conditions for a minimum. The first and second variationsare studied by methods of functional analysis.

The first chapter is devoted to the analytic formulation of the problem. Ingeneral he seeks to minimize a function

J(x)= ff(t,x,xt)dz + fo6 g(t,x)da

* Courtesy of Mathematical Reviews.

Dow

nloa

ded

12/0

6/14

to 1

30.2

16.1

29.2

08. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

352 BOOK REVIEWS

on the suitable class of n-dimensional functions

x(t) x(t, tin), G,

satisfying a set of differential constraints

h(t,x,x,) 0, v 1, ..., r < ran,

and a set of boundary conditions. He shows that most of the standard variationalproblems can be reduced to one of this type. In addition it is shown that under anassumption of positive regularity (or quasi-regularity) the problem can be putin canonical form by a Legendre transformation. Numerous examples are givento illustrate the theory and to point out difficulties that may arise.

The second chapter deals with the study of obtuse aspects of functionalanalysis which will be useful later. This includes the study of bilinear and quadraticforms on a Hilbert space. The first and second variations are studied in the thirdchapter. The theory is based on an assumption analogous to the assumption ofnormality for the classical problem of Bolza and to the assumption of control-lability for control problems. The multiplier rule is obtained by functional analysismethods. Haar’s lemma and its converse are established. The natural boundary(transversality) conditions are obtained. The second order necessary conditionand the necessary condition of Weierstrass are derived.

In the fourth chapter the general theory of the second variation is developed ina very satisfactory manner. This enables the author to establish sufficient conditionsfor a weak relative minimum. Sufficient conditions for a strong relative minimumare given in the fifth chapter. This is done by the construction of a field. In thischapter the concept of a field is carefully analyzed. After a short discussion of theCaratheodory type of field, the author develops the theory of the DeDonder-Weyl type in great detail. The last chapter is devoted to applications and extensionsof previous results, including the development of a generalized eigenvalue theory.

The book is well written and contains numerous illustrative examples. As isusually the case with an advanced book of this nature, the material will be betterunderstood if one is familiar with the standard variational theory for simpleintegral problems. The book will undoubtedly play a significant role in the futuredevelopment of multiple integral variational theory and its applications.

M. R. HESTENESUCLA

Numerical Solution of Ordinary Differential Equations. By LEON LAPIDUS andJOHN H. SEINFELD. Academic Press, New York, 1971. xii + 299 pp. $16.50.

This book is concerned with the numerical solution ofthe initial value problemfor ordinary differential equations. The authors state the following in their intro-duction: "The existence of a large number of methods, each having specialadvantages, has been a source of confusion as to what methods are best for certainclasses of problems. It is this observation which particularly motivated the writingof this book thus we have attempted to bring together all the well known classicalmethods as well as the most recent techniques in the literature in a consistent andcomprehensive framework."D

ownl

oade

d 12

/06/

14 to

130

.216

.129

.208

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php