Calculus, Vol 2, Multivariable Calculus and Linear Algebra, with Applications to Differential Equations and Probability.2nd Ed.Tom M Apostol.Wiley.1969.pdf

Tom IN. Apostol
CALCULUS
VOLUME II
Mlul ti Variable Calculus and Linear
Algebra, with Applications to
DifFeren tial Equations and Probability
SECOND EDITION
John Wiley & Sons
New York London Sydney Toronto
CONSULTING EDITOR
e Springer, Indiana University
COPYRIGHT 0 1969 BY XEROX CORPORATION.
All rights reserved. No part of the material covered by this copyright
may be produced in any form, or by any means of reproduction.
Previous edition copyright 0 1962 by Xerox Corporation.
Librar of Congress Catalog Card Number: 67-14605
ISBN 0 471 00007 8 Printed in the United States of America.
1098765432
To
Jane and Stephen
PREFACE
This book is a continuation of the author’s Calculus, Volume I, Second Edition. The
present volume has been written with the same underlying philosophy that prevailed in the
first. Sound training in technique bined with a strong theoretical development.
Every effort has been made to convey the spirit of modern mathematics without undue
emphasis on formalization. As in Volume I, historical remarks are included to give the
student a sense of participation in the evolution of ideas.
The second volume is divided into three parts, entitled Linear Analysis, Nonlinear
Ana!ysis, and Special Topics. The last two chapters of Volume I have been repeated as the
first two chapters of Volume II so that all the material on linear algebra will plete
in one volume.
Part 1 contains an introduction to linear algebra, including linear transformations,
matrices, determinants, eigenvalues, and quadratic forms. Applications are given to
analysis, in particular to the study of linear differential equations. Systems of differential
equations are treated with the help of matrix calculus. Existence and uniqueness theorems
are proved by Picard’s method of essive approximations, which is also cast in the
language of contraction operators.
Part 2 discusses the calculus of functions of several variables. Differential calculus