Introductory Mathematical Analysis

An excerpt from the PREFACE: The present course is the result of several years of study and trial in the classroom in an effort to make an introduction to college mathematics more effective, rational and better suited to its place in a scheme of education under modern conditions of life. A broader field has been attempted than is customary in books of its class. This is made possible by certain principles which controlled the construction of the text. One principle on which the course is built is correlation by topics For example, all methods of calculation have been associated in one chapter and early in the course in order to be available for use in the sequel. The function idea has also been emphasized and used as a means of correlation. Brevity and directness of treatment have contributed to reduce the size of the book. An effort has been made to keep in view of the student the steps in the development of the subject and to point out useful contacts of mathematics with affairs. The first two chapters are intended to be used for review and reference at the discretion of the instructor. Graphic representation and its uses have been given considerable attention. The simple cases of determining empirical formula give a very valuable drill in the solution of simultaneous equations and a foundation for later work in the laboratory. The treatment of the trigonometric functions is brief, direct and in some respects more advanced in style than is customary in current texts in trigonometry which are constructed mostly from the secondary school standpoint. Large use of the functions is made in a variety of applications in immediately following chapters. More than usual attention is given to vectors. The value and convenience of vector methods in science and engineering seem to justify this emphasis. The part dealing with vector products and the problems depending on it may, however, be omitted without inconvenience in later chapters. The chapter on series may seem a little heavy for freshmen but it comes in the second half of the course and is directly applied to functions within the experience of the student in the preceding text. What is given on differential and integral calculus is intended as an introduction for those who are to take the regular and fuller course in calculus. For those who are not to continue their mathematics it will furnish an introduction to the methods of calculus and some important definite applications. The integral has first been regarded as the inverse of the derivative and nothing is said about the differential. This seems natural and in accord with the idea of the solution of differential equations under many actual conditions where a function is sought whose derivative is given. Following, the integral is regarded as a summation of elements and some further applications are introduced. In the list of integrals for reference both the inverse and the differential forms are given. In general no effort at rigor beyond reasonable conviction has been attempted. Proofs have been given for some theorems that many teachers may prefer to regard as assumptions. These proofs may, therefore, be omitted at the discretion of the teacher. A number of what appear as theorems in some texts are here given as exercises. For this reason it is recommended that each student be held for practically all the exercises appearing regularly through the text. Selections may be made at the instructor's discretion from the exercises at the end of each chapter....