# The Elements of the Differential and Integral Calculus: Based on Kurzgefasstes Lehrbuch Der Differential- und Integralrechnung Von W. Nernst und A. Schönflies

D. Appleton and Company, 1900 - 410 ページ

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### 目次

 THE ELEMENTS OF ANALYTIC GEOMETRY 1 The equation of the parabola 20 Every equation of the first degree represented by a straight line 27 Concerning the nature of a general equation 35 The equation of the ellipse 42 The auxiliary circle the directrix the eccentricity 50 The form of the hyperbola 57 arT PAOB 22 Transformation of coordinates 63
 Newtons law of cooling 224 Work done in the expansion of a perfect gas at a constant 230 Maximum average temperature of a flame 236 Reactions in which the factors are only partially converted 242 DEFINITE INTEGRALS 245 The rectification of curves 267 The higher derivatives of the simplest functions 273 Oscillatory motion 280

 Exercises IX 65 Van der Waals equation 66 Polar coordinates 69 The equations of the ellipse the parabola and the hyperbola in polar coordinates 70 The spiral of Archimedes 74 Concerning imaginary points and lines 75 CHAPTER II 77 Illustrations of limits 79 Definition of limit Rigorous definition of limit 80 Application of the definition further illustrations 82 Concerning infinity 8 85 Further examples of limits 86 The fundamental theorem of limits 87 Propositions concerning limits 90 Concerning epsilons 91 Properties of epsilons 92 Exercises X 95 CHAPTER III 97 Motion on the parabola 99 Concerning speed 102 The linear expansion of a rod 105 The derivative 107 The physical signification of derivatives 109 The functionconcept 110 Exercises XI 111 i General rule for the formation of derivatives 115 Exercises XII 118 Relations between logarithms with different bases 140 Inverse trigonometric functions 147 The derivative of a power with any exponent 155 CHAPTER V 166 The integral calculus as an inverse problem 172 The physical signification of the constant of integration 180 Integration by transformation of the function to be integrated 196 Decomposition into partial fractions 203 CHAPTER VII 218
 ArT pAO 12 Eulers theorem of homogeneous functions 299 Exercises XXXIII 300 The focal properties of the parabola 301 The focal properties of the ellipse 303 The asymptotes of the hyperbola 300 301 303 306 CHAPTER X 310 The sum of infinite series 311 The geometric series 313 General theorems on the convergence of series Series with alternating signs 314 Exercises XXXIV 317 Series with varying signs 318 Series whose signs are all positive 320 Rapidity of convergency 322 Application to the series for e 323 Exercises XXXV 324 Maclaurins Theorem 325 The series for e sin x and cos x 328 Exercises XXXVI 332 Taylors Theorem 334 The logarithmic series 337 The binomial theorem 340 Exercises XXXVII 342 Table of series 346 Indeterminate forms 347 Illustrative examples of the determination of the limits of in determinate forms 350 Types of indeterminate forms 355 Exercises XXXVIII 356 Calculation with small quantities 357 Reduction with barometric readings to 0 C 358 Simplified hypsometric formula 359 MAXIMA AND MINIMA 361 Estimation of errors 383 CHAPTER XII 389 Integration 395 APPENDIX 401 著作権