Preface

PREFACE

The significance of the Calculus, the possibility of applyingit in other fields, its usefulness, ought to be kept constantlyand vividly before the student during his study of the subject,rather than be deferred to an uncertain future.

Not only for students who intend to become engineers, butalso for those planning a profound study of other sciences, theusefulness of the Calculus is universally recognized by teachers;it should be consciously realized by the student himself. It isobvious that students interested primarily in mathematics,particularly if they expect to instruct others, should recognizethe same fact.

To all these, and even to the student who expects only generalculture, the use of certain types of applications tends tomake the subject more real and tangible, and offers a basis foran interest that is not artificial. Such an interest is necessaryto secure proper attention and to insure any real grasp of theessential ideas.

For this reason, the attempt is made in this book to presentas many and as varied applications of the Calculus as it ispossible to do without venturing into technical fields whosesubject matter is itself unknown and incomprehensible to thestudent, and without abandoning an orderly presentation offundamental principles.

The same general tendency has led to the treatment oftopics with a view toward bringing out their essential usefulness.Thus the treatment of the logarithmic derivative isvitalized by its presentation as the relative rate of change of aquantity; and it is fundamentally connected with the important“ compound interest law,” which arises in any phenomenon inthe relative rate of increase (logarithmic derivative) isconstant.

Another instance of the same tendency is the attempt, in theintroduction of the precise concept of curvature, to explain thereason for the adoption of this, as opposed to other simplerbut cruder measures of bending. These are only instances, oftwo typical kinds, of the way in which the effort to bring outthe usefulness of the subject has influenced the presentation ofeven the traditional topics.

Rigorous forms of demonstration are not insisted upon, especiallywhere the precisely rigorous proofs would be beyondthe present grasp of the student. Rather the stress is laid uponthe student’s certain comprehension of that which is done, andhis conviction that the results obtained are both reasonable anduseful. At the same time, all effort has been made to avoidthose grosser errors and actual misstatements of fact whichhave often offended the teacher in texts otherwise attractiveand teachable.

Thus a proof for the formula for differentiating a logarithmis given which lays stress on the very meaning of logarithms;while it is not absolutely rigorous, it is at least just as rigorousas the more traditional proof which makes use of the limit of${(1+1/n)}^n$ as $n$ becomes infinite, and it is far more convincingand instructive. The proof used for the derivative of the sineof an angle is quite as sound as the more traditional proof(which is also indicated), and makes use of fundamentally usefulconcrete concepts connected with circular motion. Thesetwo proofs again illustrate the tendency to make the subjectvivid, tangible, and convincing to the student; this tendencywill be found to dominate, in so far as it was found possible,every phase of every topic.

Many traditional theorems are omitted or reduced in importance.In many cases, such theorems are reproduced in exercises, witha sufficient hint to enable the student to masterthem. Thus Taylor’s Theorem in several variables, for whichwide applications are not apparent until further study ofmathematics and science, is presented in this manner.

On the other hand, many theorems of importance, both frommathematical and scientific grounds, which have been omittedtraditionally, are included. Examples of this sort are the brieftreatment of simple harmonic motion, the wide application ofCavalieri’s theorem and the prismoid formula, other approximationformulas, the theory of least squares (under the headof exercises in maxima and minima), and many other topics.

The Exercises throughout are colored by the views expressedabove, to bring out the usefulness of the subject and to givetangible concrete meaning to the concepts involved. Yet formalexercises are not at all avoided, nor is this necessary if thestudent’s interest has been secured through conviction of theusefulness of the topics considered. Far more exercises arestated than should be attempted by any one student. This willlend variety, and will make possible the assignment of differentproblems to different students and to classes in successiveyears. It is urged that care be taken in selecting from theexercises, since the lists are graded so that certain groups ofexercises prepare the student for other groups which follow;but it is unnecessary that all of any group be assigned, and it isurged that in general less than half be used for any one student.Exercises that involve practical applications and othersthat involve bits of theory to be worked out by the student areof frequent occurrence. These should not be avoided, for theyare in tune with the spirit of the whole book; great care hasbeen taken to select these exercises to avoid technical conceptsstrange to the student or proofs that are too difficult.

An effort is made to remove many technical difficulties bythe intelligent use of tables. Tables of Integrals and manyother useful tables are appended; it is hoped that these willbe found usable and helpful.

Parts of the book may be omitted without destroying theessential unity of the whole. Thus the rather complete treatmentof Differential Equations (of the more elementary types)can be omitted. Even the chapter on Functions of SeveralVariables can be omitted, at least except for a few paragraphs,without vital harm; and the same may be said of the chapteron Approximations. The omission of entire chapters, of course,would only be contemplated where the pressure of time is unusual;but many paragraphs may be omitted at the discretionof the teacher.

Although care has been exercised to secure a consistent orderof topics, some teachers may desire to alter it; for example,an earlier introduction of transcendental functions and of portionsof the chapter on Approximations may be desired, and isentirely feasible. But it is urged that the comparatively earlyintroduction of Integration as a summation process be retained,since this further impresses the usefulness of the subject, andaccustoms the student to the ideas of derivative and integralbefore his attention is diverted by a variety of formal rules.

Purely destructive criticism and abandonment of coherentarrangement are just as dangerous as ultra-conservatism. Thisbook attempts to preserve the essential features of the Calculus,to give the student a thorough training in mathematical reasoning,to create in him a sure mathematical imagination, andto meet fairly the reasonable demand for enlivening and enrichingthe subject through applications at the expense of purelyformal work that contains no essential principle.

E. W. DAVIS,

W. C. BRENKE,

E. R. HEDRICK, Editor.

June, 1912.