Such gaps in understanding, if net too frequent, usually fill up of their own accord. He should not, however, let himself be disheartened by isolated difficulties which he may find in the concluding chapters. The reader will notice, too, that the general style of presentation, at first detailed, is more condensed towards the end of the book. The appendices also conrtaih some additions to the subject-matter they have been mad6 relatively concise. Instead, these are collected in appendices to the chapters, and the student whose main purpose is to acquire the facts rapidly or to proceed to practical applications may postpone reading these until he feels the need for them. The beginner should note that I have avoided blocking the entrance to the concrete facts ef the differential and integral calculus by discussions of fundamental matters, for which he is not yet ready. I do not promise to save the reader the trouble of thinking, but I do seek to lead the way straight to useful knowledge, and aim at making the subject easier to grasp, not only by giving proofs step by step, but also by throwing light on the interconnexions and purposes of the whole. The book is intended for anyone who, having passed through an ordinary course of school mathematics, wishes to apply himself to the study of mathematics or its applications to science and engineering, no matter whether he is a student of a university or technical college, a teacher, or an engineer. To me it seems extremely important that the student should be warned from the very beginning against a smug and presumptuous purism this is not the least of my purposes in writing this book. But- the attitude of "those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. Vi PREFACE TO THE FIRST GERMAN EDITION My aim is to exhibit the close connexion between analysis and its applications and, without loss of rigour and precision, to give duo credit to intuition as the source of mathematical truth. This first volume deals mainly with the integral and differential calculus for functions of one variable a second volume will be devoted to functions of several variables and some other extensions of the calc ulus. I here attempt to give it a place in the literature. With the backing of Felix Klein and others, the simultaneous treatment of differential calculus and integral calculus has steadily gained ground in lecture courses. This separation, a mere result of historical accident, with no good foundation either in theory or in practical convenience in teaching, hinders the student from grasping the central point of the calculus, namely, the connexion between definite integral, indefinite integral, and derivative. The reader will notice especially the complete break away from the out-of-date tradition of treating the differential calculus and the integral calculus separately. In order and choice of material, in fundamental aim, and perhaps also in mode of presentation, they differ considerably from the current literature. Yet I do not consider the publication of my lectures superfluous. I make no claim to have presented the beginner with the ideal textbook.
True, it is easier to perceive defects than to remedy them. He refuses to be bored by diffueeness and general statements which convey nothing to biro, and will not tolerate a pedantry which makes no distinction betweea the essential and the non-essential, and which, for the sake of a systematic set of axioms, deliberately conceals the facts to which the growth of the subject is due. PREFACE TO THE FIRST GERMAN EDITION Although there is no lack of textbooks on the differential and integral calculus, the beginner will have difficulty in finding a book that leads him straight to the heart of the subject and gives binn the power to apply it intelligently.
McSHANE Professor of Mathematics in tbc University of Virginia NEW REVISED EDITION VOLUME I COU RANT Professor of Mathematics in New York University TRANSLATED BY E.
0 kommentar(er)
