A Course of Higher Mathematics

Adiwes International Series in Mathematics, Volume 1

1st Edition - January 1, 1964

Author: V. I. Smirnov

Editor: A. J. Lohwater

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… Read more

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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.

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