A Course of Mathematical Analysis

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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