ELEMENTARY TEXTBOOK ON THE CALCULUS BY VIRGIL SNYDER, Ph. AND JOHN IRWIN HUTCHINSON, Ph.D Of Cornell University NEW YORK -:: CINCINNATI «:- CHICAGO AMERICAN BOOK COMPANY 1 THE MODERN MATHEMATICAL SERIES. Lucien Augustus Wait, (Senior Professor of Mathematics in Cornell University,) General Editor. This series includes the following works: BRIEF ANALYTIC GEOMETRY.
TaxweR and Josgrn ALLEN. ELEMENTARY ANALYTIC GEOMETRY. Tanner and Josern ALLEN. By Jamzs McMAnon and Vira ÄNYDER, INTEGRAL CALCULUS.
DIFFERENTIAL AND INTEGRAL CALCULUS. By Viram Sxypxe and J. ELEMENTARY TEXTBOOK ON THE CALCULUS. By Virain Sxypkk and J.
HIGH SCHOOL ALGEBRA. By Jamus McManon. RS 23 \ Ket Copreiant, 191%, mY AMERICAN BOOK COMPANY BL.com PREFACE Tue present volume is the outgrowth of the requirements for students in engineering and science in Cornell University, for whom a somewhat brief but adequate introduction to the Calculus is prescribed. The guiding principle in the selection and presentation of ` the topics in the following pages has been the ever increasing pressure on the present-day curriculum, especially in applied science, to limit the study of mathematics to a minimum of time and to the topics that are deemed of most immediate use to the professional course for which it is preparatory.
To what extent it is wise and justifiable to yield to this pressure it is not our purpose to discuss. But the constantly accumulating details in every pure and applied science makes this attitude a very natural one towards mathematics, as well as towards several other subjects which are subsidiary to the main object of the given course. This desire to curtail mathematical training is strikingly evidenced by the numerous recent books treating of Calculus for engineers, for chemists, or for various other professional students. Such books have no doubt served a useful purpose in various ways.
But we are of the opinion that, in spite of the unquestioned advantages of learning a new method by means of its application to a specific field, a student would ordinarily acquire too vague and inaccurate a command of the fundamental ideas of the Calculus by this one-sided presenta- tion, While a suitable illustration may clear up the difficulties 3 www.com 4 PREFACE of an abstract theory, too constant a dwelling among applica- tions alone, especially from one point of view, is quite as likely to prevent the learner from grasping the real significance of a vital principle. In recognition of the demand just referred to, we have made special effort to present the Calculus in as simple and direct a form as possible consistent with accuracy and thoroughness. Among the different features of our treatment, we may single out the following for notice. The derivative is presented rigorously as a limit.
This does not seem to be a difficult idea for the student to grasp, espe- cially when introduced by its geometrical interpretation as the slope of the line tangent to the graph of the given func- tion. For the student has already become familiar with this notion in Analytic Geometry, and will easily see that the analytic method is virtually equivalent to a particular case of the process of differentiation employed in the Calculus. In order to stimulate the student’s interest, easy applications of the Differential Calculus to maxima and minima, tangents and normals, inflexions, asymptotes, and curve tracing have been introduced as soon as the formal processes of differentia- tion have been developed. These are followed by a discussion of functions of two or more independent variables, before the more difficult subject of infinite series is introduced.
Tn the chapter on expansion, no previous knowledge of series is assumed, but conditions for convergence are discussed, and the criteria for determining the interval of convergence of those series that are usually met with in practice are derived. A chapter on the evaluation of indeterminate forms and three chapters on geometric applications furnish ample illus- www.com PREFACE 5 tration of the uses of infinite series in a wide range of problems. By reason of its significance in applications, it does not seem advisable to omit the important principle of rates. Arising out of the familiar notion of velocity, it affords an early glimpse into applications of the Calculus to Mechanics and Physics.
We do not propose to make the Calculus a treatise on Mechanics, as seems to be the tendency with some writers; but a final chapter on applications to such topies of Mechanics as are easy to comprehend at this stage is thought advisable and sufficient. Especially in treating of center of gravity, the formulas have been derived in detail, first for n particles, and then, by a limit- ing process, for a continuous mass, This was considered the more desirable, as textbooks in applied mathematics frequently lack in rigor in discussing the transition from discrete particles to a continuous mass. Besides, the derivation of these formu- las affords a very good application of the idea of the definite integral as the limit of asum. This idea has been freely and consistently used in the derivation of all applied formulas in the Integral Calculus.
However, as the formula for the length of are in polar coérdinates is especially difficult of derivation by this method, we have deduced it from the corresponding formula for rectangular codrdinates by a transformation of the variable of integration. In order to make the number of new ideas as few as possible, the notions of infinitesimals and orders of infinitesimals have been postponed to the last article on Duhamel’s principle. This principle seems to flow naturally and easily from the need of completing the proof of the formulas for center of gravity. The teacher may omit this article, but its presence should at www.com 6 PREFACE least serve the important end of calling the attention of the student to the fact that there is something yet to be done in order to make the derivations complete.
Some teachers will undoubtedly prefer to do a minimum amount of work in formal integration and use integral tables in the chapters on the applications. For such the first chapter of the Integral Calculus might suffice for drill in pure integration. The problems in this chapter are numerous, and, for the most part, quite easy, and should furnish the student a ready insight into the essential principles of integration. The characteristic features of the books on the Calculus previously published in this series have been retained.
The extensive use of these books by others, and a searching yearly test in our own classroom experience convince us that any far- reaching change could not be undertaken without endangering the merits of the book. The changes that have been made are either in the nature of a slight rearrangement, or of the addi- tion of new illustrative material, particularly in the applications. We wish to acknowledge our indebtedness to our colleagues, who have added many helpful suggestions; to Professor I. Church, of the College of Civil Engineering, for a number of very useful problems in applications of integration (See Exs.
323-324), and particu-_ larly to Professor James McMahon, who has carefully read all the manuscript, assisted throughout in the proof reading, and made many improvements in the text.com CONTENTS DIFFERENTIAL CALCULUS CHAPTER I FUNDAMENTAL PRINCIPLES ABTICLE SNE eYyr Elementary definitions. + Tilustration : Slope of a tangent to a curve ˆ. Fundamental theorems concerning limits. Continuity of fanctions.
Comparison of simultaneous increments of ‘two related variables Definition of a derivative. Process of differentiation. Differentiation of a function of a function. CHAPTER II DIFFERENTIATION oF THE EL kMENTARY Forms Differentiation of the product of a constant and a variable Differentiation of a sum +.
Differentiation of a product. Differentiation of aquotlent. * Differentiation of a commensurable power ‘of a funotion. Differentiation of implicit functions.
Elementary transcendental functions Differentiation of log, z and log, 1% : Differentiation of the simple exponential funetion Differentiation of an incommensurable power Limit of i as 6 approaches 0 ¬ Difossstisiion of sinu. Differentiation of cosu x. Differentiation of tanu. " Differentiation of sin~'w.
* Table of fundamental forma .com PAGE 15 16 17 19 21 BESSS SASSESSERES CONTENTS CHAPTER Il Successive DirrkKRENTIATION ARTICLE 25, 26. Definition of the nth derivative. Expression for the nth derivative in certain cases. CHAPTER IV Maxima AND MINIMA Increasing and decreasing functions.
- * Test for determining intervals of increasing and decreaning Turning values of afunction. Critical values of the variable. Method of determining whether #!{z) changes its sign in pees- ing through zero or infinity. ` Second method of determining whether #'œ) changes its sign in passing through zero.
The maxima and minima of any continuous ” fanctton occur alternately. Simplifications that do not alter critical values. Geometric problems in maxima and minima. « ` ° CHAPTER V Rates AND DIFFERENTIALS Rates.
Time as independent variable `. - Abbreviated notation for rates. - ` Differentials often substituted for rates =. ˆ * ` Theorem of mean value.
CHAPTER VI TDIFFERENTIAL OF AN AREA, Arc, VOLUME, AND Surrace or Revoivution Differential of an area :. Differential of an are. Trigonometric meaning of a’ re oe Differential of the volume of a surface of revolution .com PAOR 47 5 ssẽ 72 74 74 4 79 80 81 CONTENTS ARTICLE 44. Differential of a surface of revolution.
Differential of arc in polar cotrdinates. Differential of area in polar codrdinates. BERS 38 #2 § 8 CHAPTER VII ` PPLIOATIONB T0 CURVE TRACING Equation of tangent and normal : *. Length of tangent, normal, subtangent, and subnormal.
Concavity upward and downwards. : + Algebraic test for positive and negative vending Concavity and convexity toward the axis. Hyperbolic and parabolic branches. ` Definition of a rectilinear asymptote.
DETERMINATION OF ASYMPTOTES Method of limiting intercepts Method of inspection. Infinite ordinates, axymptotes poral toaxes. Method of substitution. Number of asymptotes ` “.
PoLAn COÙRDINATES de Meaning of p —. ‘ ‹ dp Relation between a and pv. * 60, Length of tangent, normal: polar eattangent, and polar sub- normal. CHAPTER VIII DIFFERENTIATION OE Functions or Two VARIABLES Definition of continuity.
ew ee Total derivative. : Differentiation of implicit functions `. Successive partial differentiation Order of differentiation indifferent : : : .com PAGE 81 $s=£šÉ&ẽ #3 104 105 106 100 110 112 116 116 118 122 122 10 CONTENTS .CHAPTER IX CHANGE OF VARIABLE ARTICLE 90. Interchange of dependent and independent variables ° + Change of the dependent variable.
Change of the independent variable. ` Simultaneous changes of dependent and of independent variables CHAPTER X EXpansion OF FUNCTIONS Convergence and divergence of series ‘. « General test for convergence. ‘ Interval of convergence.
* Remainder after 2 terms `. Maclaurin’s expansion of a fanetion in a ¬ series. ° , Form of remainder in Maclaurin’s series. Another expression for the remainder ` *.
+ * CHAPTER XI INDETERMINATE FORM8 Definition of an indeterminate form. Indeterminate forms may have determinate valass. Evaluation by development + *. : Evaluation by differentiation.
Evaluation of the indeterminate form 2. # CHAPTER XII Contact aAnp CurRVATURE Order of contact. Number of conditions implied by contact. Contact of odd and of even order Cirele of curvature.
Length of radius of curvaiers; coördinates of conter of carvatare Limiting intersection of normals + : - + ye. Direction of radius of curvatare www.com PAGE 124 125 126 126 183 138 140 141 148 150 150 168 157 158 160 161 165 167 168 169 172 172 174 175 CONTENTS 11 ARTICLE PAGE 94. Total curvature of a given arc; average curvature. Measure of curvature ata given point.
Curvature of an arc of a circle. Curvature of oeeulating cirele. Direct derivation of the expressions for.
