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(SZilt6i)(SX156i).''L., ' '\L:.*:q!fl& -'Advanced Calculus.0Fifth Edition.I~3it* i[ # ] Wilfred Kaplan8.Preface.to the Fifth EditionAs I fecall, it was in 1948 that Mark Morkovin, a colleague in engineering, approached me to suggest that I write a text for engineering students needing to proceed.beyond elementary calculus to handle the new applications of mathematics. World.War I1 had indeed created many new demands for mathematical skills in a variety.of fieldsMark was persuasive and I prepared a book of 265 pages, which appeared in.lithoprinted form, and it was used as the text for a new course for third-year studentsThe typesetting was done using a "varityper," a new typewriter that had keys for.mathematical symbolsIn the summer of 1949 I left Ann Arbor for a sabbatical year abroad, and.we rented our home to a friend and colleague Eric Reissner, who had a visiting.appointment at the University of Michigan. Eric was an adviser to a new publisher,.Addison-Wesley, and learned about my lithoprinted book when he was asked to.teach a course using it. He wrote to me, asking that I consider having it published.by Addison-WesleyThus began the course of this book. For the first edition the typesetting was.carried out with lead type and I was invited to watch the process. It was impressive to.see how the type representing the square root of a function was created by physically.cutting away at an appropriate type showing the square root sign and squeezing type.for the function into it. How the skilled person carrying this out would have marveled.at the computer methods for printing such symbols!.This edition differs from the previous one in that the chapter on ordinary differential.equations included in the third edition but omitted in the fourth edition has been.restored as Chapter 9. Thus the present book includes all the material present in the.previous editions, with the exception of the introductory review chapter of the first.editionA number of minor changes have been made throughout, especially some updating of the referencesThe purpose of including all the topics is to make the book more useful for.reference. Thus it can serve both as text for one or more courses and as a source of.information after the courses have been completed.ABOUT.THE BOOK.The background assumed is that usually obtained in the freshman-sophomore calculus sequence. Linear algebra is not assumed to be known but is developed in the first.chapter. Subjects discussed include all the topics usually found in texts on advanced.calculus. However. there is more than the usual emphasis on applications and on.physical motivation. Vectors are introduced at the outset and serve at many points.to indicate geometrical and physical significance of mathematical relationsNumerical methods are touched upon at various points, both because of their.practical value and because of the insights they give into the theory. A sound level.of rigor is maintained throughout. Definitions are clearly labeled as such and all.important results are formulated as theorems. A few of the finer points of real variable.theory are treated at the ends of Chapters 2, 4, and 6. A large number of problems.(with answers) are distributed throughout the text. These include simple exercises.as well as complex ones planned to stimulate critical reading. Some -points of the.theory are relegated to the problems. with hints given where appropriate. Generous.references to the literature are given, and each chapter concludes with a list of books.for supplementary reading. Starred sections are less essential in a first course.Chapter 1 opens with a review of "ectors in space. determinants, and linear equations, and then develops matrix algebra, including Gaussian elimination, and.n-dimensional geometry, with stress on linear mappin.gs.The second chapter takes up.partial derivatives and develops them with the aid of vectors (gradient, for example).and matrices;