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Infinitesimal Analysis BERKELEY > LIBRARY UNIVERSITY OF fc CALIFORNIA. MATH/STAT LIBRARY MATH/STAT LIBRARY MATH/STAT, INFINITESIMAL ANALYSIS. INFINITESIMAL ANALYSIS BY WILLIAM BENJAMIN SMITH PROFESSOR OF MATHEMATICS IN TULANE UNIVERSITY VOL. I. ELEMENTARY: REAL VARIABLES JJeto fork THE MACMILLAN COMPANY LONDON : MACMILLAN AND CO., LIMITED 1898 All riyhts reserved GLASGOW: PRINTED AT THE UNIVERSITY BY ROBERT MACLKHOSE AND CO. PREFACE. THIS volume has been written on what appeared, in the light of ten years experience in teaching the Calculus, to be lines of least resistance. The aim has been, within a prescribed expense of time and energy, to penetrate as far as possible, and in as many directions, into the subject in hand, that the student should attain as wide knowledge of the matter, as full comprehension of the methods, and as clear consciousness of the spirit and power of this analysis as the nature of the case would admit. Accordingly, what seemed to be natural suggestions and impulses towards near-lying extensions or generaliza tions have often been followed, and even allowed to direct the course of the discussion. Hereby, necessarily, the exposition has suffered in symmetry and in systematic character but has the defects of its own ; everything qualities. The aim already stated, no less than the plan of its pursuit, has excluded Weierstrassian rigor from many investigations and compelled the postponement of im portant discussions as too subtle for an early stage of in no has been to study ; particular, attempt made deal with Series, unless the most familiar, or to follow in the wake of the masters of e-methods. But real difficulties have not been knowingly disguised, and the reader is vi PREFACE. often warned that the treatment given is only provisional and must await further precision or delimitation. It has also resulted that some topics have been merely mentioned, it appearing at once that further investigation would lead straightway beyond the prescribed limits of the volume, or into difficulties for which the reader was not prepared. But it is believed that the glimpses thus afforded of wider and higher fields have a distinct value, and may attract the student with promise of richer reward for renewed effort. The book is, in fact, written for such as feel genuine interest in the subject, whose minds are open to such inspiration. It was in the original scheme of this volume to add an Appendix, in which certain purely algebraic assump tions of the text, as the Exponental Theorem and Decom into should be position Part-Fractions, carefully grounded ; but the volume having grown beyond expectation, such addenda have been omitted, as in any case dispensable. Though the discussion has been confined to real variables an occasional passing employment of the Eulerian i has seemed unavoidable. The illustrations and exercises have been chosen with frequent reference to practical or theoretic importance, or to historic interest, and without any pretence of novelty. The author acknowledges in full his great indebtedness to the works of Amstein, Byerly, Edwards, Greenhill, Gregory, Harnack, Johnson, Sohncke, Stolz, Todhunter, Williamson, and others, from which the examples, in par ticular, have been mostly culled. He desires also to return thanks publicly to Professor Dr. Carlo Veneziani for patient and painstaking revision both of the MS. and of the proof-sheets. Such services would under circumstances have been valuable but any ; PREFACE. vii under the peculiar embarrassments that have beset every stage of the composition of this volume they have been invaluable. Thanks are also due to the author s colleague, Mr. J. E. Lombard, for assistance in proof-reading, and especially in drawing the figures. The author will be grateful for indication of errors, either of pen or of type. If, in spite of all such, the book shall prove useful, whether by enlightening or by inciting, and shall advance the mastery of the most powerful weapon of thought yet devised by the wit of man, its end will be in some measure attained. W. B. S. NEW ORLEANS, Christmas, 1897. CONTENTS. CHAPTER I. FUNDAMENTAL NOTIONS AND OPERATIONS. ARTICLES PAGES - 1, 2. The Notion of Limit : Illustrations 1 3-5. Definitions 2, 3 6-11. Theorems on Limits - - - 3-5 - 12, 13. Definition, Notation, and Classification of Functions 5, 6 - 14, 15. Correspondences, Continuity 7, 8 - 16-18. Difference-Quotient : Illustrations 9, 10 19-22. Derivative and Differential Coefficient - 11, 12 23-29. Theorems on Derivatives - 12-15 - 30, 31. Geometrical Illustration. Exceptional Cases - 15, 16 32. Problem of Tangents - 17 33, 34. Illustration from Mechanics - 18, 19 35, 36. Further Considerations 19, 20 37-41. Elementary Derivations - 21-23 42-44. Geometric Proof, Ratio of Chord and Arc - - 24-26 45. Other modes of Proof - - 27 46. Other Trigonometric Functions 28 47. Sense of Derivative - 28 48, 49. Inverse Functions - 28, 29 Exercises - 30-32 50, 51. Tangent, Normal, etc. - 32 52. Derivative of the Arc - 33 53. Polar Coordinates 34 54. Infinitesimals of Higher Order 36 55. Additional Formulae - 37 56. Angle between two Curves - 38 57-59. Pedal Curves - - 38 - - - Exercises - - - 39, 40 x CONTENTS. ARTICLES PAGES 60. Pencils and Parameters 41 61. Intersection of Elements 42 62. Intersection of Consecutives - 43 - 44 63, 64. Envelopes 43, 65. Parameters with Conditions - - 45, 46 Exercises 47 66-68. Higher Derivatives - 48-50 69. Theorem of Leibnitz - 51 ax nx 70, 71. Derivatives of e and e X 52 72. Meaning of Operator - 53 73. Application to Sine and Cosine 53 74. Extension of Leibnitz Theorem 54 - 54 75, 76. Operator xD - 77, 78. Anti-operators 55, 56 79. Indefiniteness of Inverse Operations 57 Exercises .-..-. 58-60 CHAPTER II. INTEGRATION. 80, 81. Problem of Areas, Illustration - 61, 62 82. Integration not Summation - 63 83. Deductions - 64, 65 84. Theorem of Mean Value 66 85. Derivation of Integral as to Extremes 67 86-90. Various Quadratures - 68-73 91-94. Need of better Method. Integration and Derivation 73-75 95. Definite and Indefinite Integrals - 76 96. Fundamental Forms - - 77-78 97. Change of Variable - 79 98. Integration by Parts Illustrations - 80-81 99. Circle and Equiaxial Hyperbola - 82 100. Anti-functions Ee-defined - 83 101, 102. Hyperbolic Cosine and Sine 84 103. Anti-hyper-functions 85 104. The Arguments pure Numbers 86 105. Relations of the Functions - 86 106. Exponentals - 87 107-109. Other Hyper-functions. Gudermannians. Geome tric Property - 88-90 110. Ellipse and Hyperbola 91 111. Logarithmic Expression - 91 CONTENTS. xi ARTICLES PAGES 112. Imaginary Exponentals - 92 113. Reduction to Standard Forms 92 Exercises - 93 114, 115. Powers of Sine and Cosine, Secant and Cosecant 93, 94 116, 117. Odd and Even Powers 95 118-121. Tangent and Special Integrals - 96,97 122. Simple Fractions - 97 123. Rational Algebraic Functions 98 124. Irrational Quadratics 99, 100 125. Higher Functions - 101 126. Exponentals and Trigonometric Functions 102 127. Illustrations and Exercises - 103-112 CHAPTER III. APPLICATIONS. 128, 129. Bernoulli s Series. Taylor s Series - - 113, 114 130. Lagrange s and Cauchy s Forms - 114 131. Interpretation of the Series 115 132, 133. Stirling s Series. Relations of the Series - 116 134, 135. Development of Logarithm - - 117, 118 136,137. Expansion of Anti-tangent - 119 138. Binomial Theorem 120 139. Newton s Method -. 120 140, 141. Bernoulli s Numbers 121 Exercises - - 122, 123 EVANESCENTS. 142-144. Removal of Factors - - 124, 125 145. Illustrations - 126 146. Application of Taylor s Series 127 147. Simplifications 128 148, 149. Other Illusory Forms 128 150. Illustrations - - 129-131 Exercises - - 132-134 MAXIMA AND MINIMA. 151, 152. Definitions. Relativity 134 153, 154. Conditions - 135 155. Continued Vanishing of Derivatives 136 156. Use of Taylor s Theorem - 137 157. Geometric Interpretation - 138 xii CONTENTS. ARTICLES PAGES 158. Rule for Practice - 139 139 159, 160. Exceptional Oases. Discontinuities 161. Geometric Problems - 140 141 162. Simplifications Exercises _..--- 141-148 CHAPTER IV. GEOMETRIC INTERPRETATION OF HIGHER DERIVATIVES. 163. Order of Contact - 149 164. Tangency of Curves - 150 165. Concavity and Convexity - 151 - 166, 167. Curvature -152,153 168-170. Uniform Curvature, Standard of Curvature, Centre of Curvature - 154-156 - 171, 1 72. Evolute and Involute 157, 158 173. Centre of Curvature redefined 159 - 174, 175. Higher and Closest Contact 159-161 176, 177. Exceptional Points. Maximal and Minimal Curva ture - 162, 163 178. Osculatory Circle - 164 179. Application to the Conic - 164 180. Illustrations - - - - - 165-170 CHAPTER V. VARIOUS PROBLEMS OF INTEGRATION. 181. Quadrature - - 171, 172 182. Rectification - 173 183. Cubature - 173, 174 184. Quadrature of Revolutes - 175, 176 185, 186. Averages. Mass-centre - 177-179 187, 188. Moments - 180 - - 189. Radius of Gyration . 181 190, 191. Parallel Axes. Normal Axes - 181, 182 192. Density 182 193, 194. Theorems of Pappus - 183 195, 196. Choice of Elements. Symmetry - 184 197. Illustrations ------ 185-191 CONTENTS. Xlll CHAPTER VI. PARTIAL DERIVATION. ARTICLES PAGES 198. Double Dependence - 192 199. Geometric Depiction 193 200. Parallel Sections 193 201. Tangency 194 202. Partial Derivatives - 195 203. Equation of Tangent Plane - 196 204. Equations of Normal . 197 205. Simultaneous Changes 197, 198 199 206, 207. Theorem of Total Differential. Limitations 207*. Tortuous Curves 200-207 208. Use of Third Variable 207 209 209, 210. Higher Derivatives. Formation 208, - 210-212 211, 212. Taylor s Theorem extended. Application - 213 213, 214. Osculating Paraboloid, Indicatrix 215. Illustration - 214 215-217 216,217. Curvature. Solid Angles 218-221 218, 219. Maxima and Minima 220. Illustrations - 222-234 CHANGE OF VARIABLES. 234 221, 222. Use of a Third Independent Variable 235-237 223, 224. Eeversion. Eeciprocants 225. Transformation from System to System 237, 238 226. Change of Independents 239 227. Jacobians Extension of notion 240 228. Derivation through Jacobians 240 229. Vanishing of Jacobians 241 243 230, 231.
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