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.
Author(s): Edwin Bidwell Wilson
Edition: 1
Publisher: Ginn and company
Year: 1911
Language: English
Pages: 581
City: Boston, New York
Tags: Mathematical Analysis
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
