A Course of Mathematical Analysis

A Course of Mathematical Analysis

International Series of Monographs on Pure and Applied Mathematics

1st Edition - January 1, 1963

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  • Author: A. F. Bermant
  • eBook ISBN: 9781483137322

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Description

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.

Table of Contents


  • 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

Product details

  • No. of pages: 508
  • Language: English
  • Copyright: © Pergamon 1963
  • Published: January 1, 1963
  • Imprint: Pergamon
  • eBook ISBN: 9781483137322

About the Author

A. F. Bermant

About the Editors

I. N. Sneddon

S. Ulam

M. Stark

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