Advanced Calculus: A Text Upon Select Parts of Differential Calculus, Differential Equations, Integral Calculus, Theory of Functions, With Numerous Exercises

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An introduction to partial differentiation, complex numbers and vectors, differential equations, infinite integrals, calculus of variations, infinite series, functions of complex variable, elliptic functions & Integrals, functions of real variables, and other important topics in advanced mathematics. C O N T E N T S INTRODUCTORY REVIEW CHAPTER I REVIEW OF FUNDAMENTAL RULES SECTION 1. On differentiation . . . . . . . . . . 1 4. Logarithmic, exponential, and hyperbolic functions . . . 4 6. Geometric properties of the derivative . . . . . . 7 8. Derivatives of higher order . . . . . . . . 11 10. The indefinite integral . . . . . . . . . 15 13. Aids to integration . . . . . . . . . . 18 16. Definite integrals . . . . . . . . . . 24 CHAPTER II REVIEW OF FUNDAMENTAL THEORY 18. Numbers and limits . . . . . . . . . 33 21. Theorems on limits and on sets of points . . . . . 37 23. Real functions of a real variable . . . . . . . 40 26. The derivative . . . . . . . . . . 45 28. Summation and integration . . . . . . . . 50 PART I. DIFFERENTIAL CALCULUS CHAPTER III TAYLOR’S FORMULA AND ALLIED TOPICS 31. Taylor's Formula . . . . . . . . . . 55 33. Indeterminate forms, infinitesimals, infinities . . . . . 81 36. Infinitesimal-analysis . . . . . . . . . 68 40. Some differential geometry . . . . . . . . 78 CHAPTER IV _ PARTIAL DIFFERENTIATION; EXPLICIT FUNCTIONS 43. Functions of two or more variables . . . . . . 87 46. First partial derivatives . . . . . . . . . 93 50. Derivatives of higher order . . . . . . . . 102 54. Taylor’s Formula and applications . . . . . . . 112 CHAPTER V PARTIAL DIFFERENTIATION; IMPLICIT FUNCTIONS 56. The simplest case; F(X,Y) = 0 . . . . . . . 117 59. More general cases of implicit functions . . . . . 122 62. Functional determinants or Jacobians . . . . . . 129 65. Envelopes of curves and surfaces . . . . . . . 135 68. More differential geometry . . . . . . . . 143 CHAPTER VI COMPLEX NUMBERS AND VECTORS 70. Operators and operations . . . . . . . . 149 71. Complex numbers . . . . . . . . . . 153 73. Functions of a complex variable . . . . . . . 157 75. Vector sums and products . . . . . . . . 163 77. Vector differentiation . . . . . . . . . 170 PART II. DIFFERENTIAL EQUATIONS CHAPTER VII GENERAL INTRODUCTION TO DIFFERENTIAL EQUATIONS 81. Some geometric problems . . . . . . . . 179 83. Problems in mechanics and physics . . . . . . 184 85. Linear element and differential equation . . . . . 191 87. The higher derivatives; analytic approximations . . . . 197 CHAPTER VIII THE COMMONER ORDINARY DIFFERENTIAL EQUATIONS 89. Integration by separating the variables . . . . . . 203 91. Integrating factors . . . . . . . . . 207 95. Linear equations with constant coefficients . . . . . 214 98. Simultaneous linear equations with constant coefficients . . 223 CONTENTS vii CHAPTER IX ADDITIONAL TYPES OF ORDINARY EQUATIONS 100. Equations of the first order and higher degree . . . . 228 102. Equations of higher order . . . . . . . . 234 104. Linear differential equations . . . . . . . 240 107. The cylinder functions . . . . . . . . . 247 CHAPTER X DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES 109. Total differential equations . . . . . . . . 254 111. Systems of simultaneous equations . . . . . . 260 113. Introduction to partial differential equations . . . . 267 116. Types of partial differential equations . . . . . . 273 PART III. INTEGRAL CALCULUS CHAPTER XI 0N SIMPLE INTEGRALS 118. Integrals containing a parameter . . . . . . . 281 121. Curvilinear or line integrals . . . . . . . . 288 124. Independency of the path . . . . . . . . 298 127. Some critical comments . . . . . . . . 308 CHAPTER XII ON MULTIPLE INTEGRALS 129. Double sums and double integrals . . . . . . 315 133. Triple integrals and change of variable . . . . . 326 135. Average values and higher integrals . . . . . . 332 137. Surfaces and surface integrals . . . . . . . 338 CHAPTER XIII ON INFINITE INTEGRALS 140. Convergence and divergence . . . . . . . . 352 142. The evaluation of infinite integrals . . . . . . 360 144. Functions defined by infinite integrals . . . . . . 368 CHAPTER XIV SPECIAL FUNCTIONS DEFINED BY INTEGRALS 147. The Gamma and Beta functions . . . . . . . 378 150. The error function . . . . . . . . . 386 153. Bessel functions . . . . . . . . . . 393 CHAPTER XV THE CALCULUS OF VARIATIONS 155. The treatment of the simplest case . . . . . . 400 157. Variable limits and constrained minima . . . . . . 404 159. Some generalizations . . . . . . . . . . 409 PART IV. THEORY OF FUNCTIONS CHAPTER XVI INFINITE SERIES 162. Convergence or divergence of series . . . . . . 419 165. Series of functions . . . . . . . . . 430 168. Manipulation of series . . . . . . . . . 440 CHAPTER XVII SPECIAL INFINITE DEVELOPMENTS 171. The trigonometric functions . . . . . . . . 453 173. Trigonometric or Fourier series . . . . . . . 458 175. The Theta functions . . . . . . . . . 467 9 CHAPTER XVIII FUNCTIONS OF A COMPLEX VARIABLE 178. General theorems . . . . . . . . . . 476 180. Characterization of some functions . . . . . . 482 183. Conformal representation . . . . . . . . 490 185. Integrals and their inversion . . . . . . . . 496 CHAPTER XIX ELLIPTIC FUNCTIONS AND INTEGRALS 187. Legendre's integrals I and its inversion . . . . . . 503 190. Legendre's integrals II and III . . . . . . . 511 192. Weierstrass's integrals and its inversion . . . . . . 517 CHAPTER XX . FUNCTIONS OF REAL VARIABLES 194. Partial differential equations of physics . . . . . 624 196. Harmonic functions; general theorems . . . . . 530 198. Harmonic functions; special theorems . . . . . . 537 201. The potential integrals . . . . . . . . . 546 BOOK LIST . . . . . . . . . . . 565 INDEX- . . . . . . . . . . . . 657