A Course of Higher Mathematics

A Course of Higher Mathematics

Adiwes International Series in Mathematics, Volume 1

1st Edition - January 1, 1964

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  • Author: V. I. Smirnov
  • eBook ISBN: 9781483156361

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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 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.

Table of Contents

  • 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


    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


    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


    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


    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


    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



Product details

  • No. of pages: 558
  • Language: English
  • Copyright: © Pergamon 1964
  • Published: January 1, 1964
  • Imprint: Pergamon
  • eBook ISBN: 9781483156361

About the Author

V. I. Smirnov

About the Editor

A. J. Lohwater

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