We also give the Preface to Courant's English edition at this link. |

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 him the power to apply it intelligently. He refuses to be bored by diffuseness and general statements which convey nothing to him, and will not tolerate a pedantry which makes no distinction between 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.

True, it is easier to perceive defects than to remedy them. I make no claim to have presented the beginner with the ideal textbook. Yet I do not consider the publication of my lectures superfluous. In order and choice of material, in fundamental aim, and perhaps also in mode of presentation, they differ considerably from the current literature.

The reader will notice especially the complete break away from the out-of-date tradition of treating the differential calculus and the integral calculus separately. 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 connection between definite integral, indefinite integral, and derivative. With the backing of Felix Klein and others, the simultaneous treatment of differential calculus and integral calculus has steadily gained ground in lecture courses. I here attempt to give it a place in the literature. 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 calculus.

My aim is to exhibit the close connection between analysis and its applications and, without loss of rigour and precision, to give due credit to intuition as the source of mathematical truth. 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. 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. 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.

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. 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 inter-connections and purposes of the whole.

The beginner should note that I have avoided blocking the entrance to the concrete facts of the differential and integral calculus by discussions of fundamental matters, for which be is not yet ready. 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 appendices also contain some additions to the subject-matter; they have been made relatively concise. The reader will notice, too, that the general style of presentation, at first detailed, is more condensed towards the end of the book. He should not, however, let himself be disheartened by isolated difficulties which he may find in the concluding chapters. Such gaps in understanding, if not too frequent, usually fill up of their own accord.

R Courant

JOC/EFR November 2006

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