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Advanced calculus

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Advanced calculus

[ M ] Wilfred Kaplan

Advanced calculus ll, 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 pr

oceed beyond elementary calculus to handle the new applications of mathematics. World War II had indeed created many new demands for mathematical skil Advanced calculus

ls in a variety of fields.Mark was persuasive and I prepared a book of 265 pages, which appeared in lithoprinted form, and it was used as the text for

Advanced calculus

a new course for third-year students. The typesetting was done using a “varityper,” a new typewriter that had keys for mathematical symbols.In the su

[ M ] Wilfred Kaplan

Advanced calculus ment 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 t

o teach a course using it. He wrote to me, asking that I consider having it published by Addison-Wesley.Thus began the course of this book. For the fi Advanced calculus

rst 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

Advanced calculus

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 die fu

[ M ] Wilfred Kaplan

Advanced calculus is EditionThis 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 excep Advanced calculus

tion of the introductory review chapter of the first edition.A number of minor changes have been made throughout, especially some updating of the refe

Advanced calculus

rences.The 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 a

[ M ] Wilfred Kaplan

Advanced calculus omore calculus sequence. Linear algebra is not assumed to be known but is developed in the first chapter. Subjects discussed include all the topics us

ually found in texts on advanced calculus. However, there is more than the usual emphasis on applications and on physical motivation. Vectors are intr Advanced calculus

oduced at the outset and serve at many points to indicate geometrical and physical significance of mathematical relations.Numerical methods are touche

Advanced calculus

d 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 maint

[ M ] Wilfred Kaplan

Advanced calculus able 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 incl

ude simple exercises as well as complex ones planned to stimulate critical reading. Some-points of the theory arc relegated to the problems, with hint Advanced calculus

s given where appropriate. Generous references to the literature are given, and each chapter concludes with a list of books for supplementary reading.

Advanced calculus

Starred sections are less essential in a first course.Topical SummaryChapter I opens with a review of vectors in space, determinants, and linear equa

[ M ] Wilfred Kaplan

Advanced calculus r takes up partial derivatives and develops them w ith the aid of vectors (gradient, for example) and matrices; partial derivatives are applied to geo

metry and to maximum-minimum problems. The third chapter introduces divergence and curl and the basic identities; orthogonal coordinates are treated c Advanced calculus

oncisely; final sections provide an introduction to tensors in rt-dimcnsional space.The fourth chapter, on integration, reviews definite and indefinit

Advanced calculus

e integrals, using numerical methods to show how the latter can be constructed; multiple integrals arc treated carefully, with emphasis on the rule fo

[ M ] Wilfred Kaplan

Advanced calculus hese is completed at the end of Chapter 6, where they arc• 2 ■www.pdfgrip.comrelated to infinite series. Chapter 5 is devoted to line and surface inte

grals. Although the notions are first presented without vectors, it very soon becomes clear how natural the vector approach is for this subject. Line Advanced calculus

integrals are used to provide an exceptionally complete treatment of transformation of variables in a double integral. Many physical applications, inc

Advanced calculus

luding potential theory, arc given.Chapter 6studies infinite series without assumption of previous knowledge. The notions of upper and lower limits ar

[ M ] Wilfred Kaplan

Advanced calculus rticular, the root test. With its aid, the treatment of power series is greatly simplified. Uniform convergence is presented with great care and appli

ed to power series. Final sections point out the parallel with improper integrals; in particular, power scries are shown to correspond to the Laplace Advanced calculus

transform.Chapter 7is a complete treatment of Fourier scries at an elementary level. The first sections give a simple introduction with many examples;

Advanced calculus

the approach is gradually deepened and a convergence theorem is proved. Orthogonal functions are then studied, with the aid of inner product, norm, a

[ M ] Wilfred Kaplan

Advanced calculus a corollary. Closing sections cover Bessel functions. Fourier integrals, and generalized functions.Chapter 8develops the theory of analytic functions

with emphasis on power scries. Laurent scries and residues, and their applications. It also provides a full treatment of conformal mapping, with many Advanced calculus

examples and physical applications and extensive discussion of the Dirichlet problem.Chapter 9assumes some background in ordinary differential equati

Advanced calculus

ons. Linear systems are treated with the aid of matrices and applied to vibration problems, power series methods are treated concisely. A unified proc

[ M ] Wilfred Kaplan

Advanced calculus lays great stress on the relationship between the problem of forced vibrations of a spring (or a system of springs) and the partial differential equat

ionf)U„ + huf — k2V2u = i'(x. y. z, t).By pursuing this idea vigorously the discussion uncovers the physical meaning of the partial differential equat Advanced calculus

ion and makes the mathematical tools used become natural. Numerical methods arc also motivated on a physical basis.Throughout, a number of references

Advanced calculus

arc made to the text Calculus and Linear Algebra by Wilfred Kaplan and Donald J. Lewis (2 vols.. New York. John Wiley &. Sons, 1970-1971), cited simpl

[ M ] Wilfred Kaplan

Advanced calculus with a knowledge of only the simplest notions of the previous ones. The later portions of the chapter may depend on some of the later portions of earl

ier ones. Il is thus possible to construct a course using just the earlier portions of several chapters. The following is an illustration of a plan fo Advanced calculus

r a one-semester course, meeting four hours• 3 •www.pdfgrip.com

[ M ] Wilfred Kaplan