A Course of Higher Mathematics
Adiwes International Series in Mathematics, Volume 1
- 1st Edition - January 1, 1964
- Imprint: Pergamon
- Author: V. I. Smirnov
- Editor: A. J. Lohwater
- Language: English
- Hardback ISBN:9 7 8 - 0 - 0 8 - 0 1 0 2 0 6 - 1
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 2 3 9 4 - 3
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 5 6 3 6 - 1
A Course of Higher Mathematics, I: Elementary Calculus is a five-volume course of higher mathematics used by mathematicians, physicists, and engineers in the U.S.S.R. This volume… Read more
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Request a sales quoteA Course of Higher Mathematics, I: Elementary Calculus is a five-volume course of higher mathematics used by mathematicians, physicists, and engineers in the U.S.S.R. This volume deals with calculus and principles of mathematical analysis including topics on functions of single and multiple variables. The functional relationships, theory of limits, and the concept of differentiation, whether as theories and applications, are discussed. This book also examines the applications of differential calculus to geometry. For example, the equations to determine the differential of arc or the parameters of a curve are shown. This text then notes the basic problems involving integral calculus, particularly regarding indefinite integrals and their properties. The application of definite integrals in the calculation of area of a sector, the length of arc, and the calculation of the volumes of solids of a given cross-section are explained. This book further discusses the basic theory of infinite series, applications to approximate evaluations, Taylor's formula, and its extension. Finally, the geometrical approach to the concept of a number is reviewed. This text is suitable for physicists, engineers, mathematicians, and students in higher mathematics.
Introduction
Prefaces to Eighth and Sixteenth Russian Editions
Chapter I. Functional Relationships and the Theory of Limits
§ 1. Variables
1. Magnitude and Its Measurement
2. Number
3. Constants and VariableS
4. Interval
5. The concept of function
6. The Analytic Method of Representing Functional Relationships
7. Implicit Functions
8. The Tabular Method
9. The Graphical Method of Representing Numbers
10. Coordinates
11. Graphs. The Equation of a Curve
12. Linear Functions
13. Increment. The Basic Property of a Linear Function
14. Graph of Uniform Motion
15. Empirical Formula
16. Parabola of the Second Degree
17. Parabola of the Third Degree
18. The Law of Inverse Proportionality
19. Power Functions
20. Inverse Functions
21. Many-valued Functions
22. Exponential and Logarithmic Functions
23. Trigonometric Functions
24. Inverse Trigonometric, or Circular, Functions
§ 2. The Theory of Limits. Continuous Functions
25. Ordered Variables
26. Infinitesimals
27. The Limit of a Variable
28. Basic Theorems
29. Infinitely Large Magnitudes
30. Monotonic Variables
31. Cauchy's Test for the Existence of a Limit
32. Simultaneous Variation of Two Variables, Connected by a Functional Relationship
33. Example
34. Continuity of Functions
35. The Properties of Continuous Functions
36. Comparison of Infinitesimals and of Infinitely Large Magnitudes
37. Examples
38. The Number E
39. Unproved Hypotheses
40. Real Numbers
41. The Operations on Real Numbers
42. The Strict Bounds of Numerical Sets. Tests for the Existence of a Limit
43. Properties of Continuous Functions
44. Continuity of Elementary Functions
Exercises
Chapter II. Differentiation: Theory and Applications
§ 3. Derivatives and Differentials of the First Order
45. The Concept of Derivative
46. Geometrical Significance of the Derivative
47. Derivatives of some Simple Functions
48. Derivatives of Functions of a Function, and of Inverse Functions
49.Table of Derivatives, and Examples
50. The Concept of Differential
51. Some Differential Equations
52. Estimation of Errors
§ 4. Derivatives and Differentials of Higher Orders
53. Derivatives of Higher Orders
54. Mechanical Significance of the Second Derivative
55. Differentials of Higher Orders
56. Finite Differences of Functions
§ 5· Application of Derivatives to the Study of Functions
57. Tests for Increasing and Decreasing Functions
58. Maxima and Minima of Functions
59. Curve Tracing
60. The Greatest and Least Values of a Function
61. Format's Theorem
62. Rolle's Theorem
63. Lagrange's Formula
64. Cauchy's Formula
65. Evaluating Indeterminate Forms
66. Other Indeterminate Forms
§ 6· Functions of Two Variables
67. Basic Concepts
68. The Partial Derivatives and Total Differential of a Function of two Independent Variables
69. Derivatives of Functions of a Function and of Implicit Functions
§ 7· Some Geometrical Applications of the Differential Calculus
70. The Differential of Arc
71. Concavity, Convexity, and Curvature
72. Asymptotes
73. Curve-tracing
74. The Parameters of a Curve
75. Van der Waal's Equation
76. Singular Points of Curves
77. Elements of a Curve
78. The Catenary
79. The Cycloid
80. Epicycloid and Hypocycloid
81. Involute of a Circle
82. Curves in Polar Coordinates
83. Spirals
84. The Limaçon and Cardioid
85. Cassini's Ovals and the Lemniscate
Exercises
Chapter III. Integration: Theory and Applications
§ 8. Basic Problems of the Integral Calculus. The Indefinite Integral
86. The Concept of an Indefinite Integral
87. The Definite Integral as the Limit of a Sum
88. The Relation between the Definite and Indefinite Integrals
89. Properties of Indefinite Integrals
90. Table of Elementary Integrals
91. Integration by Parts
92. Rule for Change of Variables. Examples
93. Examples of Differential Equations of the First Order
§ 9. Properties of the Definite Integral
94. Basic Properties of the Definite Integral
95. Mean Value Theorem
96. Existence of the Primitives
97. Discontinuities of the Integrand
98. Infinite Limits
99. Change of Variable for Definite Integrals
100. Integration by Parts
§ 10. Applications of Definite Integrals
101. Calculation of Area
102. Area of a Sector
103. Length of Arc
104. Calculation of the Volumes of Solids of Given Cross-section
105. Volume of a Solid of Revolution
106. Surface Area of a Solid of Revolution
107. Determination of Center of Gravity. Guldin's Theorem
108. Approximate Evaluation of Definite Integrals. Beetangle and Trapezoid Formula
109. Tangent Formula, and Poncelet's Formula
110. Simpson's Formula
111. Evaluation of Definite Integrals with Variable Upper Limits
112. Graphical Methods
113. Areas under Rapidly Oscillating Curves
§ 11. Further Remarks on Definite Integrals
114. Preliminary Concepts
115. Darboux's Theorem
116. Functions Integrable in Riemann's Sense
117. Properties of Integrable Functions
Exercises
Chapter IV. Series. Applications to Approximate Evaluations
§ 12. Basic Theory of Infinite Series
118. Infinite Series
119. Basic Properties of Infinite Series
120. Series with Positive Terms. Tests for Convergence
121. Cauchy's and d'Alembert's Tests
122. Cauchy's Integral Test for Convergence
123. Alternating Series
124. Absolutely Convergent Series
125. General Test for Convergence
§ 13. Taylor's Formula and Its Applications
126. Taylor's Formula
127. Different Forms of Taylor's Formula
128. Taylor and Maclaurin Series
129. Expansion of Ex
130. Expansion of Sin x and Cos x
131. Newton's Binomial Expansion
132. Expansion of Log (1+x)
133. Expansion of Arc Tan x
134. Approximate Formula
135. Maxima, Minima, and Points of Inflexion
136. Evaluation of Indeterminate Forms
§ 14. Further Remarks on the Theory of Series
137. Properties of Absolutely Convergent Series
138. Multiplication of Absolutely Convergent Series
139. Kummor's Test
140. Gauss's Test
141. Hypergeometric Series
142. Double Series
143. Series with Variable Terms. Uniformly Convergent Series
144. Uniformly Convergent Sequences of Functions
145. Properties of Uniformly Convergent Sequences
146. Properties of Uniformly Convergent Series
147. Tests for Uniform Convergence
148. Power Series. Radius of Convergence
149. Abel's Second Theorem
150. Differentiation and Integration of Power Series
Exercises
Chapter V. Functions of Several Variables
§ 15. Derivatives and Differentials
151. Basic Concepts
152. Passing to a Limit
153. Partial Derivatives and Total Differentials of the First Order
154. Euler's Theorem
155. Partial Derivatives of Higher Orders
156. Differentials of Higher Orders
157. Implicit Functions
158. Example
159. Existence of Implicit Functions
160. Curves in Space and Surfaces
§ 16. Taylor's Formula. Maxima and Minima of Functions of Several Variables
161. Extension of Taylor's Formula to Functions of Several Independent Variables
162. Necessary Conditions for Maxima and Minima of Functions
163. Investigation of the Maxima and Minima of a Function of Two Independent Variables
164. Examples
165. Additional Remarks on Finding the Maxima and Minima of a Function
166. The Greatest and Least Values of a Function
167.Conditional mMxima and Minima
168. Supplementary Remarks
169. Examples
Exercises
Chapter VI. Complex Numbers. The Foundations of Higher Algebra. Integration of Various Functions
§ 17. Complex Numbers
170. Complex Numbers
171. Addition and Subtraction of Complex Numbers
172. Multiplication of Complex Numbers
173. Division of Complex Numbers
174. Raising to a Power
175. Extraction of Roots
176. Exponential Functions
177. Trigonometric and Hyperbolic Functions
178. The Catenary
179. Logarithms
180. Sinusoidal Quantities and Vector Diagrams
181. Examples
182. Curves in the Complex Form
183. Representation of Harmonic Oscillations in Complex Form
§ 18. Basic Properties and Evaluation of the Zeros of Integral Polynomials
184. Algebraic Equations
185. Factorization of Polynomials
186. Multiple Zeros
187. Horner's Rule
188. Highest Common Factor
189. Real Polynomials
190. The Relationship between the Roots of an Equation and Its Coefficients
191. Equations of the Third Degree
192. The Trigonometric Form of Solution of Cubic Equations
193. The Method of Successive Approximations
194. Newton's Method
195. The Method of Simple Interpolation
§ 19. Integration of Various Functions
196. Reduction of Rational Fractions to Partial Fractions
197. Integration of Rational Fractions
198. Integration of Expressions Containing Radicals
199. Integrals of the Type f R(x, Vax2+bx+c)dx
200. Integrals of the Form f R(sin x,cos x)dx
201. Integrals of the form f eax[P(x)cos bx+Q(x)sin bx]dx
Answers
Index
- Edition: 1
- Published: January 1, 1964
- No. of pages (eBook): 558
- Imprint: Pergamon
- Language: English
- Hardback ISBN: 9780080102061
- Paperback ISBN: 9781483123943
- eBook ISBN: 9781483156361
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