# A Course of Higher Mathematics

## 1st Edition

### Adiwes International Series in Mathematics

**Authors:**V. I. Smirnov

**Editors:**A. J. Lohwater

**eBook ISBN:**9781483185088

**Imprint:**Pergamon

**Published Date:**1st January 1964

**Page Count:**644

## Description

A Course of Higher Mathematics, Volume II: Advanced Calculus covers the theory of functions of real variable in advanced calculus.

This volume is divided into seven chapters and begins with a full discussion of the solution of ordinary differential equations with many applications to the treatment of physical problems. This topic is followed by an account of the properties of multiple integrals and of line integrals, with a valuable section on the theory of measurable sets and of multiple integrals. The subsequent chapters deal with the mathematics necessary to the examination of problems in classical field theories in vector algebra and vector analysis and the elements of differential geometry in three-dimensional space. The final chapters explore the Fourier series and the solution of the partial differential equations of classical mathematical physics.

This book will prove useful to advanced mathematics students, engineers, and physicists.

## Table of Contents

Introduction

Preface to the Sixth Edition

Preface to the Fourteenth Edition

Chapter I Ordinary Differential Equations

§ 1. Equations of the First Order

1. General Principles

2. Equations with Separable Variables

3. Homogeneous Equations

4. Linear Equations; Bernoulli's Equation

5. Finding the Solution of a Differential Equation with a given Initial Condition

6. The Euler—Cauchy Method

7. The General Solution

8. Clairaut's Equation

9. Lagrangian Equations

10. The Envelope of a Family of Curves, and Singular Solutions

11. Equations Quadratic in y'

12. Isogonal Trajectories

§ 2. Differential Equations of Higher Orders; Systems of Equations

13. General Principles

14. Graphical Methods of Integrating Second Order Differential Equations

15. The Equation y(n) = ƒ (X)

16. Bending of a Beam

17. Lowering the Order of a Differential Equation

18. Systems of Ordinary Differential Equations

19. Examples

20. Systems of Equations and Equations of Higher Orders

21. Linear Partial Differential Equations

22. Geometrical Interpretation

23. Examples

Chapter II Linear Differential Equations. Supplementary Remarks on the Theory of Differential Equations

§ 3. General Theory; Equations with Constant Coefficients

24. Linear Homogeneous Equations of the Second Order

25. Non-Homogeneous Linear Equations of the Second Order

26. Linear Equations of Higher Orders

27. Homogeneous Equations of the Second Order with Constant Coefficients

28. Non-Homogeneous Linear Equations of the Second Order with Constant Coefficients

29. Particular Cases

30. Linear Equations of Higher Orders with Constant Coefficients

31. Linear Equations and Oscillatory Phenomena

32. Free and Forced Oscillations

33. Sinusoidal External Forces and Resonance

34. Impulsive External Forces

35. Statical External Forces

36. The Strength of a Thin Elastic Rod, Compressed by Longitudinal Forces (Euler's Problem)

37. Rotating Shaft

38. Symbolic Method

39. Linear Homogeneous Equations of Higher Orders with Constant Coefficients

40. Linear Non-Homogeneous Equations with Constant Coefficients

41. Example

42. Euler's Equation

43. Systems of Linear Equations with Constant Coefficients

44. Examples

§ 4. Integration with the Aid of Power Series

45. Integration of a Linear Equation, Using a Power Series

46. Examples

47. Expansion of Solutions into Generalized Power Series

48. BessePs Equation

49. Equations Reducible to Bessel's Equation

§ 5. Supplementary Notes on the Theory of Differential Equations

50. The Method of Successive Approximations for Linear Equations

51. The Case of a Non-Linear Equation

52. Singular Points of First Order Differential Equations

53. The Streamlines of Collinear Plane Fluid Motion

Chapter III Multiple and Line Integrals. Improper Integrals that Depend on a Parameter

§ 6. Multiple Integrals

54. Volumes

55. Double Integrals

56. Evaluation of Double Integrals

57. Curvilinear Coordinates

58. Triple Integrals

59. Cylindrical and Spherical Coordinates

60. Curvilinear Coordinates in Space

61. Basic Properties of Multiple Integrals

62. Surface Areas

63. Integrals Over a Surface and Ostrogradskii's Formula

64. Integrals Over a given Side of a Surface

65. Moments

§ 7. Line Integrals

66. Definition of a Line Integral

67. Work Done by a Field of Force. Examples

68. Areas and Line Integrals

69. Green's Formula

70. Stokes' Formula

71. Independence of a Line Integral on the Path in a Plane

72. Multiply Connected Domains

73. Independence of a Line Integral on the Path in Space

74. Steady-State Flow of Fluids

75. Integrating Factors

76. Exact Differential Equations in the Case of Three Variables

77. Change of Variables in Double Integrals

§ 8. Improper Integrals and Integrals that Depend on a Parameter

78. Integration Under the Integral Sign

79. Dirichlet's Formula

80. Differentiation Under the Integral Sign

81. Examples

82. Improper Integrals

83. Conditionally Convergent Integrals

84. Uniformly Convergent Integrals

85. Examples

86. Improper Multiple Integrals

87. Examples

§ 9. Supplementary Remarks on the Theory of Multiple Integrals

88. Preliminary Concepts

89. Basic Theorems Regarding Sets

90. Interior and Exterior Areas

91. Measurable Sets

92. Independence on the Choice of Axes

93. The Case of any Number of Dimensions

94. Darboux's Theorem

95. Integrable Functions

96. Properties of Integrable Functions

97. Evaluation of Double Integrals

98. n-Tuple Integrals

99. Examples

Chapter IV Vector Analysis and Field Theory

§ 10. Basic Vector Algebra

100. Addition and Subtraction of Vectors

101. Multiplication of a Vector by a Scalar. Coplanar Vectors

102. Resolution of a Vector into Three Non-Coplanar Components

103. Scalar Product

104. Vector Products

105. The Relationship between Scalar and Vector Products

106. The Velocities at Points of a Rotating Rigid Body; the Moment of a Vector

§ 11. Field Theory

107. Differentiation of Vectors

108. Scalar Field and Gradient

109. Vector Fields. Curl and Divergence

110. Lamellar and Solenoidal Fields

111. Directed Elementary Areas

112. Some Formulae of Vector Analysis

113. Motion of a Rigid Body and Small Deformations

114. Equation of Continuity

115. Hydrodynamical Equations for an Ideal Fluid

116. Equations of Sound Propagation

117. Equation of Thermal Conduction

118. Maxwell's Equations

119. Laplace's Operator in Orthogonal Coordinates

120. Differentiation in the Case of a Variable Field

Chapter V Foundations of Differential Geometry

§ 12. Curves on a Plane and in Space

121. The Curvature of a Plane Curve; the Evolute

122. Involutes

123. The Natural Equation of a Curve

124. The Fundamental Elements of Curves in Space

125. Frenet's Formula

126. The Osculating Plane

127. The Helix

128. Field of Unit Vectors

§ 13. Elementary Theory of Surfaces

129. The Parametric Equations of a Surface

130. Gauss First Differential Form

131. Gauss Second Differential Form

132. The Curvature of Lines Ruled on Surfaces

133. Dupin's Indicatrix and Euler's Formula

134. Finding the Principal Radii of Curvature and Principal Directions

135. Line of Curvature

136. Dupin's Theorem

137. Examples

138. Gaussian Curvature

139. The Variation of an Elementary Area and the mean Curvature

140. Envelopes of Surfaces and Curves

141. Developable Surfaces

Chapter VI Fourier Series

§ 14. Harmonic Analysis

142. Orthogonality of the Trigonometric Functions

143. Diriehlet's Theorem

144. Examples

145. Expansion in the Interval (0, π)

146. Periodic Functions of Period 2l

147. Average Error

148. General Orthogonal Systems of Functions

149. Practical Harmonic Analysis

§ 15. Supplementary Remarks on the Theory of Fourier Series

150. Expansion in Fourier Series

151. Second mean Value Theorem

152. Dirichlet Integrals

153. Diriehlet's Theorem

154. Polynomial Approximations to Continuous Functions

155. The Closure Equation

156. Properties of Closed Systems of Functions

157. The Character of the Convergence of Fourier Series

158. Improving the Convergence of Fourier Series

159. Example

§ 16. Fourier Integrals and Multiple Fourier Series

160. Fourier's Formula

161. Fourier Series in Complex Form

162. Multiple Fourier Series

Chapter VII The Partial Differential Equations of Mathematical Physics

§ 17. The Wave Equation

163. The Equation of the Vibration of a String

164. D'Alembert's Solution

165. Particular Cases

166. Finite String

167. Fourier's Method

168. Harmonics and Standing Waves

169. Forced Vibrations

170. Concentrated Force

171. Poisson's Formula

172. Cylindrical Waves

173. The Case of n-Dimensional Space

174. Non-Homogeneous Wave Equation

175. Point Sources

176. The Transverse Vibrations of a Membrane

177. Rectangular Membrane

178. Circular Membranes

179. The Uniqueness Theorem

180. Applications of Fourier Integrals

§ 18. The Equation of Telegraphy

181. Fundamental Equations

182. Steady-State Processes

183. Transient Processes

184. Examples

185. Generalized Equation of Vibration of a String

186. The General Case of an Infinite Circuit

187. Fourier's Method for a Finite Circuit

188. The Generalized Wave Equation

§ 19. The Vibrations of Rods

189. Fundamental Equations

190. Particular Solutions

191. The Expansion of an Arbitrary Function

§ 20. Laplace's Equation

192. Harmonic Functions

193. Green's Formula

194. The Fundamental Properties of Harmonic Functions

195. The Solution of Dirichlet's Problem for a Circle

196. Poisson Integrals

197. Dirichlet's Problem for a Sphere

198. Green's Function

199. The Case of a Halfspace

200. Potential of a Distributed Mass

201. Poisson's Equation

202. Kirchhoff's Formula

§ 21. The Equation of Thermal Conduction

203. Fundamental Equations

204. Infinite Rod

205. Semi-Infinite Rods

206. Rods Bounded at both Ends

207. Supplementary Remarks

208. The Case of a Sphere

209. The Uniqueness Theorem

Index

## Details

- No. of pages:
- 644

- Language:
- English

- Copyright:
- © Pergamon 1964

- Published:
- 1st January 1964

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483185088