A Course of Mathematical Analysis, Part I is a textbook that shows the procedure for carrying out the various operations of mathematical analysis. Propositions are given with a precise statement of the conditions in which they hold, along with complete proofs. Topics covered include the concept of function and methods of specifying functions, as well as limits, derivatives, and differentials. Definite and indefinite integrals, curves, and numerical, functional, and power series are also discussed.
This book is comprised of nine chapters and begins with an overview of mathematical analysis and its meaning, together with some historical notes and the geometrical interpretation of numbers. The reader is then introduced to functions and methods of specifying them; notation for and classification of functions; and elementary investigation of functions. Subsequent chapters focus on limits and rules for passage to the limit; the concepts of derivatives and differentials in differential calculus; definite and indefinite integrals and applications of integrals; and numerical, functional, and power series.
This monograph will be a valuable resource for engineers, mathematicians, and students of engineering and mathematics.
Author(s): A. F. Bermant
Series: International Series of Monographs on Pure and Applied Mathematics
Edition: 1
Publisher: Pergamon Press
Year: 1963
Language: English
Pages: 510
City: Oxford
Tags: Mathematical Analysis, Calculus
Preface to the Seventh Edition
Introduction
1. Mathematical Analysis and Its Meaning
1. "Elementary" and "Higher" Mathematics
2. Magnitudes. Variables and Functional Relationships
3. Mathematical Analysis and Reality
2. Some Historical Notes
4. Great Russian Mathematicians: L. P. Euler, N. I . Lobachevskii, P. L. Chebyshev
5. Leading Russian Applied Mathematicians: N. E. Zhukovskii, S. A. Chaplygin, A. N. Krylov
3. Real Numbers
6. Real Numbers. The Real Axis
7. Intervals. Absolute Values
8. a Note on Approximations
Chapter I Functions
1. Functions and Methods of Specifying Them
9. The Concept of Function
10. Methods of Specifying Functions
2. Notation for and Classification of Functions
11. Notation
12. Function of a Function. Elementary Functions
13. The Classification of Functions
3. Elementary Investigation of Functions
14. Domain of Definition of a Function. Domain of Definiteness of an Analytic Expression
15. Elements of the Behavior of Functions
16. Graphical Investigation of a Function. Linear Combinations of Functions
4. Elementary Functions
17. Direct Proportionality and Linear Functions. Increments
18. Quadratic Functions
19. Inverse Proportionality and Linear Rational Functions
5. Inverse Functions. Power, Exponential and Logarithmic Functions
20. The Concept of Inverse Function
21. Power Functions
22. Exponential and Hyperbolic Functions
23. Logarithmic Functions
6. Trigonometric and Inverse Trigonometric Functions
24. Trigonometric Functions
25. Simple and Compound Harmonic Vibrations
26. Inverse Trigonometric Functions
Chapter II Limits
1. Basic Definitions
27. The Limit of a Function of an Integral Argument
28. Examples
29. The Limit of a Function of a Continuous Argument
2. Non-Finite Magnitudes. Rules for Passage to the Limit
30. Infinitely Large Magnitudes. Bounded Functions
31. Infinitesimals
32. Rules for Passage to the Limit
33. Examples
34. Tests for the Existence of a Limit
3. Continuous Functions
35. Continuity of a Function
36. Points of Discontinuity of a Function
37. General Properties of Continuous Functions
38. Operations on Continuous Functions. Continuity of the Elementary Functions
4. Comparison of Infinitesimals. Some Important Limits
39. Comparison of Infinitesimals. Equivalent Infinitesimals
40. Examples of Ratios of Infinitesimals
41. The Number e. Natural Logarithms
Chapter III Derivatives and Differentials. The Differential Calculus
1. The Concept of Derivative. Rate of Change of a Function
42. Some Physical Concepts
43. Derivative of a Function
44. Geometrical Interpretation of Derivative
45. Some Properties of the Parabola
2. Differentiation of Functions
46. Differentiation of the Results of Arithmetical Operations
47. Differentiation of a Function of a Function
48. Derivatives of the Basic Elementary Functions
49. Logarithmic Differentiation. Differentiation of Inverse and Implicit Functions
50. Graphical Differentiation
3. Differentials. Differentiability of a Function
51. Differentials and Their Geometrical Interpretation
52. Properties of the Differential
53. Application of the Differential to Approximations
54. Differentiability of a Function. Smoothness of a Curve
4. Derivative as Rate of Change (Further Examples)
55. Rate of Change of a Function with Respect to a Function. Parametric Specification of Functions and Curves
56. Rate of Change of Radius Vector
57. Rate of Change of Length of Arc
58. Processes of Organic Growth
5. Repeated Differentiation
59. Derivatives of Higher Orders
60. Leibniz's Formula
61. Differentials of Higher Orders
Chapter IV The Investigation of Functions and Curves
1. The Behavior of a Function "at a Point"
62. Construction of a Graph from "Elements" 197
63. Behavior of a Function "at a Point". Extrema
64. Tests for the Behavior of a Function "at a Point"
2. Applications of the First Derivative
65. Theorems of Rolle and Lagrange
66. Application of Lagrange's Formula to Approximations
67. Behavior of a Function in an Interval
68. Examples
69. a Property of the Primitive
3. Applications of the Second Derivative
70. Second Sufficient Test for an Extremum
71. Convexity and Concavity of a Curve. Points of Inflexion
72. Examples
4. Auxiliary Problems. Solution of Equations
73. Cauchy's Theorem and L'Hôpital's Rule
74. Asymptotic Variation of Functions and the Asymptotes of Curves
75. General Scheme for Investigation of Functions. Examples
76. Solution of Equations. Multiple Roots
5. Taylor's Formula and Its Applications
77. Taylor's Formula for Polynomials
78. Taylor's Formula
79. Some Applications of Taylor's Formula. Examples
6. Curvature
80. Curvature
81. Radius, Center and Circle of Curvature
82. Evolute and Involute
83. Examples
Chapter V The Definite Integral
1. The Definite Integral
84. Area of a Curvilinear Trapezoid
85. Examples From Physics
86. The Definite Integral. Existence Theorem
87. Evaluation of the Definite Integral
2. Basic Properties of the Definite Integral
88. Elementary Properties of the Definite Integral
89. Change of Direction and Subdivision of the Interval of Integration. Geometrical Interpretation of the Integral
90. Estimation of the Definite Integral
3. Basic Properties of the Definite Integral (Continued). The Newton-Leibniz Formula
91. Mean Value Theorem. Mean Value of a Function
92. Derivative of an Integral with Respect to Its Upper Limit
93. The Newton-Leibniz Formula
Chapter VI The Indefinite Integral. The Integral Calculus
1. The Indefinite Integral and Indefinite Integration
94. The Indefinite Integral. Basic Table of Integrals
95. Elementary Rules for Integration
96. Examples
2. Basic Methods of Integration
97. Integration by Parts
98. Change of Variable
3. Basic Classes of Integrable Functions
99. Linear Rational Functions
100. Examples
101. Ostrogradskii's Method
102. Some Irrational Functions
103. Trigonometric Functions
104. Rational Functions of x and √ax2+bx+c
105. General Remarks
Chapter VII Methods of Evaluating Definite Integrals. Improper Integrals
1. Methods of Evaluating Integrals
106. Definite Integration by Parts
107. Change of Variable in a Definite Integral
2. Approximate Methods
108. Numerical Integration
109. Graphical Integration
3. Improper Integrals
110. Integrals with Infinite Limits
111. Tests for Convergence and Divergence of Integrals with Infinite Limits
112. Integral of a Function with Infinite Jumps
113. Tests for Convergence and Divergence of Integrals of Discontinuous Functions
Chapter VIII Applications of the Integral
1. Elementary Problems and Methods of Solution
114. Method of "Summation of Elements"
115. Method of "Differential Equation". Scheme for Solution of Problems
116. Examples
2. Some Problems of Geometry and Statics. Processes of Organic Growth
117. Area of a Figure
118. Length of Arc
119. Volume of a Body
120. Area of Surface of Revolution
121. Center of Gravity and Guldin's Theorems
122. Processes of Organic Growth
Chapter IX Series
1. Numerical Series
123. Series. Convergence
124. Series with Positive Terms. Sufficient Tests for Convergence
125. Series with Arbitrary Terms. Absolute Convergence
126. Operations on Series
2. Functional Series
127. Definitions. Uniform Convergence
128. Integration and Differentiation of Functional Series
3. Power Series
129. Taylor's Series
130. Examples
131. Interval and Radius of Convergence
132. General Properties of Power Series
4. Power Series (Continued)
133. Another Method of Expanding Functions in Taylor's Series
134. Some Applications of Taylor's Series
135. Functions of a Complex Variable. Euler's Formula
Index
