Calculus

Preface IT IS the purpose of this book to set forth in a systematic and thorough manner the fundamental principles, methods, and uses of calculus. The presentation is designed to give the student a good understanding of the wide range of applications of calculus in science and engineering, to make him aware of the logical structure of the subject, and to train him in the techniques of formulating and solving problems. In pursuit of these broad objectives this revised edition is written in the same spirit as the original edition. The book has been extensively rewritten, with the principal intention of providing an abundance of instructive and interesting exercises to assist the student in mastering each topic as it is introduced. We have taken particular care to see that the earlier exercises in each set are free from unnecessary algebraic or trigonometric complications. The student is thus free to concentrate all his attention on the formulation of the problem and on the essential principles of calculus involved in the solution. The texts of many sections have either been completely rewritten, or have been amplified by the addition of more illustrative examples to clarify the exposition at points where classroom experience has shown that fuller explanations are helpful. Approximately forty new figures have been added. One of the foremost problems confronting the teacher of calculus is that of presenting the subject of limits successfully. It is not enough to rely entirely on the student's intuitive grasp of the limit concept, important as this is. Intuitive understanding of limit processes, as they are met in the everyday situations of geometry and physics, should be carefully cultivated. But the student should also be guided by the laying down of' sufficiently precise definitions and theorems to make it clear that the method of limits is systematic, and that its development is based upon logical arguments from specific hypotheses. Most teachers will agree that proofs of theorems on limits should not be required of beginning students. It is important, however, if the methods of analysis are to be properly understood, that the student be permitted to read, at an early stage, some of the theorems and proofs which are most fundamental. The theorems on limits of sums, products, and quotients are presented in Chapter I, §5, and their uses are illustrated. Proofs are deferred until the end of the chapter (§9), and may well be omitted from the formal part of the course. A very little of the refined arithmetical treatment of limits is needed in the elementary stages of calculus. It is necessary, however, to have available a method for asserting the existence of a limit in certain situations. We have chosen the Cauchy criterion for the existence of a limit as fundamental, and announced it without proof (Chapter XIV). The fact that a bounded, nondecreasing sequence is convergent is then derived. The discussion of these matters occupies a brief chapter immediately before the chapter on infinite series. The existence of the limit defining the base of natural logarithms is treated separately, in an appendix. A feature of the present edition is the early introduction of the inverse of differentiation in Chapter IV. Discussion there is limited to powers of x, and the application is to problems in rectilinear motion, that is, determination of the motion from knowledge of the acceleration or velocity together with initial conditions. The inverse of differentiation is studied at greater length in Chapter VIII, and some simple but important differential equations are considered. The definite integral is defined as the limit of approximating sums, and the connection between differentiation and integration is worked out analytically. Not until this has been done is the word integration used in connection with the inverse of differentiation. Adherence to this procedure in treating integration seems to us to be important. The existence of the definite integral of a continuous function is not proved, but the necessity of a proof is discussed, and the relation of this matter to the definition of area is considered. The formulation of certain geometrical and physical quantities as definite integrals is made precise through the use of what we have termed Duhamel's Principle. This is enunciated without the use of the term infinitesimal, in a form consistent with the method of presentation of the definite integral. The essential form of the principle, as here stated, has been given by the late Professor Osgood. One unusual feature of the book is that it contains a full chapter devoted to the exposition of solid analytic geometry. This is particularly designed to lay a foundation for the treatment of partial differentiation and multiple integration. It may be omitted entirely if the student's preparation on the topics of the chapter is based upon a separate treatment of the subject. A small amount of material dealing with the use of rectangular coordinates in describing figures in three-dimensional space is included in Chapter XI (§88), to assist the student in the application of integration to the finding of volumes (by slicing). We take this opportunity of thanking our colleagues for their cooperation and for numerous helpful suggestions. Also, we acknowledge our debt to our students, from whom we have learned much, and from whom we shall doubtless learn more. G.E.F.S. A. E. T.