# A Course of Higher Mathematics

### Adiwes International Series in Mathematics, Volume 4

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A Course of Higher Mathematics, Volume IV provides information pertinent to the theory of the differential equations of mathematical physics. This book discusses the application of mathematics to the analysis and elucidation of physical problems. Organized into four chapters, this volume begins with an overview of the theory of integral equations and of the calculus of variations which together play a significant role in the discussion of the boundary value problems of mathematical physics. This text then examines the basic theory of partial differential equations and of systems of equations in which characteristics play a key role. Other chapters consider the theory of first order equations. This book discusses as well some concrete problems that indicate the aims and ideas of the calculus of variations. The final chapter deals with the boundary value problems of mathematical physics. This book is a valuable resource for mathematicians and readers who are embarking on the study of functional analysis.

## Table of Contents

Introduction

Preface to the Second Edition

Preface to the Third Edition

Chapter 1 Integral Equations

1. Examples of the Formation of Integral Equations

2. The Classification of Integral Equations

3. Orthogonal Systems of Functions

4. Fredholm Equations of the Second Kind

5. Method of Successive Approximations and the Resolvent

6. Existence and Uniqueness Theorem

7. Fredholm's Determinant

8. Fredholm's Equation for Any λ

9. Adjoint Integral Equation

10. The Case of an Eigenvalue

11· Fredholm Minors

12. Degenerate Equations

13. Examples

14. Generalization of the Results Obtained

15. The Selection Principle

16. The Selection Principle (Continued)

17. Unbounded Kernels

18. Integral Equations with Unbounded Kernels

19. The Case of an Eigenvalue

20. Equations with Continuous Iterated Kernels

21. Symmetric Kernels

22. Expansion in Eigenfunctions

23. Dini's Theorem

24. Expansion of Iterated Kernels

25. Classification of Symmetric Kernels

26. Extremal Properties of the Eigenfunctions

27. Mercer's Theorem

28. The Case of a Weakly Polar Kernel

29. Non-Homogeneous Equations

30. Fredholm's Treatment in the Case of a Symmetric Kernel

31. Hermitian Kernels

32. Equations Reducible to Symmetric Equations

33. Examples

34. Kernels Depending on a Parameter

35. Space of Continuous Functions

36. Linear Operators

37. Existence of the Eigenvalue

38. Sequences of Eigenvalues and Expansion Theorem

39. Space of Complex Continuous Functions

40. Completely Continuous Integral Operators

41. Normal Operators

42. The Case of Functions of Several Variables

43. Volterra's Equation

44. Laplace Transformation

45. Convolution of Functions

46. Volterra Equation of Special Type

47. Volterra Equation of the First Kind

48. Examples

49. Weighted Integral Equations

50. Integral Equation of the First Kind with Cauchy Kernels

51. Boundary Value Problems for Analytic Functions

52. Integral Equations of the Second Kind with Cauchy Kernels

53. Boundary Value Problems for the Case of a Segment

54. Inversion of a Cauchy Type Integral

55. Fourier's Integral Equation

56. Equations in the Case of an Infinite Interval

57. Examples

58. The Case of a Semi-Infinite Interval

59. Examples

60. More General Equations

Chapter II The Calculus of Variations

61. Statement of the Problem

62. Fundamental Lemmas

63. Euler's Equation in the Elementary Case

64. The Case of Several Functions and Higher Order Derivatives

65. The Case of Multiple Integrals

66. Remarks on the Euler and Ostrogradskii Equations

67. Examples

68. Isoperimetric Problems

69. Conditional Extremum

70. Examples

71. Invariance of the Euler and Ostrogradskii Equations

72. Parametric Forms

73. Geodesies in n-Dimensional Space

74. Natural Boundary Conditions

75. Functionals of a More General Type

76. General Form of the First Variation

77. Transversality Condition

78. Canonical Variables

79. Field of Extremals in Threedimensional Space

80. Theory of Fields in the General Case

81. A Singular Case

82. Jacobi's Theorem

83. Discontinuous Solutions

84. One-Sided Extrema

85. Second Variation

86. Jacobi's Condition

87. Weak and Strong Extrema

88. Weierstrass's Function

89. Examples

90. The Ostrogradskii—Hamilton Principle

91. Principle of Least Action

92. Strings and Membranes

93. Rods and Plates

94. the Fundamental Equations of the Theory of Elasticity

95. Absolute Extrema

96. Absolute Extrema (Continued)

97. Direct Methods of the Calculus of Variations

98. Examples

Chapter III Fundamental Theory of Partial Differential Equations

§ 1. First Order Equations

99. Linear Equations with Two Independent Variables

100. Cauchy's Problem and Characteristics

101. The Case of Any Number of Variables

102. Examples

103. Auxiliary Theorem

104.Non-Linear First Order Equations

105. Characteristic Manifolds

106. Cauchy's Method

107. Cauchy's Problem

108. Uniqueness of the Solution

109. The Singular Case

110. Any Number of Independent Variables

111. Complete, General and Singular Integrals

112. the Complete Integral and Cauchy's Problem

113. Examples

114. The Case of Any Number of Variables

115. Jacobi's Theorem

116. Systems of Two First Order Equations

117. The Lagrange-Charpit Method

118. Systems of Linear Equations

119. Complete and Jacobian Systems

120. Integration of Complete Systems

121. Poisson Brackets

122. Jacobi's Method

123. Canonical Systems

124. Examples

125. The Method of Majorant Series

126. Kovalevskaya's Theorem

127. Equations of Higher Order

§ 2. Equations of Higher Orders

128. Types of Second Order Equation

129. Equations with Constant Coefficients

130. Normal Forms with Two Independent Variables

131. Cauchy's Problem

132. Characteristic Strips

133. Higher Order Derivatives

134. Real and Imaginary Characteristics

135. Fundamental Theorems

136. Intermediate Integrals

137. The Monge-Ampere Equations

138. Characteristics with Any Number of Independent Variables

139. Bicharacteristics

140. The Connection with Variational Problems

141. The Propagation of a Surface of Discontinuity

142. Strong Discontinuities

143. Riemann's Method

144. Characteristic Initial Data

145. Existence Theorems

146. Method of Successive Approximations

147. Green's Formula

148. Sobolev's Formula

149. Sobolev's Formula (Continued)

150. Construction of the Function A

151. The General Case of Initial Data

152. Generalized Wave Equation

153. The Case of Any Number of Independent Variables

154. Basic Inequalities

155. Theorems on the Uniqueness and Continuous Dependence of the Solutions

156. The Case of the Wave Equation

157. Supplementary Propositions

158. Generalized Solutions of the Wave Equation

159. Equations of the Elliptic Type

160. Generalized Solution of Poisson's Equation

§ 3. Systems of Equations

161. Characteristics of Systems of Equations

162. Kinematic Compatibility Conditions

163. Dynamic Compatibility Conditions

164. The Equations of Hydrodynamics

165. Equations of the Theory of Elasticity

166. Anisotropie Elastic Media

167. Electromagnetic Waves

168. Strong Discontinuities in the Theory of Elasticity

169. Characteristics and Higher Frequencies

170. The Case of Two Independent Variables

171. Examples

Chapter IV Boundary Value Problems

§ 1. Boundary Value Problems for an Ordinary Differential Equation

172. Green's Function for a Linear Second Order Equation

173. Reduction to an Integral Equation

174. Symmetry of Green's Function

175. The Eigenvalues and Eigenfunctions of a Boundary Value Problem

176. The Signs of the Eigenvalues

177. Examples

178. The Generalized Green's Function

179. Legendre Polynomials

180. Hermite and Laguerre Functions

181. Equations of the Fourth Order

182. Steklov's Stricter Expansion Theorems

183. The Justification of Fourier's Method for the Equation of Heat Conduction

184. The Justification of Fourier's Method for the Equations of Vibrations

185. Uniqueness Theorem

186. Extremal Properties of the Eigenvalues and Eigenfunctions

187. Courant's Theorem

188. Asymptotic Expression for the Eigenvalues

189. Asymptotic Expression for the Eigenfunctions

190. Ritz's Method

191. Ritz's Example

§ 2. Equations of the Elliptic Type

192. The Newtonian Potential

193. The Potential of a Double Layer

194. Properties of the Potential of a Simple Layer

195. The Normal Derivative of the Potential of a Simple Layer

196. The Normal Derivative of the Potential of a Simple Layer (Continued)

197. The Direct Value of the Normal Derivative

198. The Derivative of the Potential of a Simple Layer with Respect to Any Direction

199. Logarithmic Potential

200. Integral Formulae and Parallel Surfaces

201. Sequences of Harmonic Functions

202. Formulation of Interior Boundary Value Problems for Laplace's Equation

203. The Exterior Problem in the Case of a Plane

204. Kelvin's Transformation

205. Uniqueness of the Solution of Neumann's Problem

206. The Solution of Boundary Value Problems in the Three-Dimensional Case

207. Investigation of the Integral Equations

208. Summary of the Results Relating to the Solution of Boundary Value Problems

209. Boundary Value Problems on a Plane

210. An Integral Equation for Spherical Functions

211. The Heat Equilibrium of a Radiating Body

212. Schwartz's Method

213. Proof of the Lemma

214. Schwartz's Method (Continued)

215. Sub- and Superharmonic Functions

216. Auxiliary Propositions

217. The Method of Lower and Upper Functions

218. Investigation of Boundary Values

219. Laplace's Equation in n-Dimensional Space

220. Green's Function for the Laplace Operator

221. Properties of Green's Function

222. Green's Function in the Case of a Plane

223. Examples

224. Green's Function and the Non-Homogeneous Equation

225. Eigenvalues and Eigenfunctions

226. The Normal Derivative of an Eigenfunction

227. Extremal Properties of the Eigenvalues and Eigenfunctions

228. Helmholtz's Principle and the Radiation Principle

229. Uniqueness Theorem

230. the Principle of Limiting Amplitude and the Principle of Limiting Absorption

231. Boundary Value Problems for Helmholtz's Equation

232. Diffraction of Electromagnetic Waves

233. The Magnetic Intensity Vector

234. The Uniqueness of the Solution of Dirichlet's Problem for Elliptic Equations

235. The Equation Δv — λv = 0

236. An Asymptotic Expression for the Eigenvalues

237. Proof of the Auxiliary Theorem

238. Linear Equations of a More General Type

239. Linear Elliptic Equations of the Second Order

240. Green's Tensor

241. The Plane Statical Problem of the Theory of Elasticity

§ 3. Equations of the Parabolic and Hyperbolic Type

242. the Dependence of the Solutions of the Heat Conduction Equation on the Initial and Boundary Conditions and the Function ƒ

243. Potentials for the Heat Conduction Equation in the One-Dimensional Case

244. Heat Sources in the Multi-Dimensional Case

245. Green's Function for the Heat Conduction Equation

246. Application of Laplace Transforms

247. Application of Finite Differences

248. Fourier's Method

249. Non-Homogeneous Equations

250. Properties of the Solutions of the Heat Conduction Equation

251. The Generalized Potentials of a Simple and Double Layer in the One-Dimensional Case

252. Sub- and Superparabolic Functions

253. Fundamental Inequalities for Solutions of the Wave Equation

254. The Case of the Nonhomogeneous Equation

255. Fourier's Method and Generalized Solutions

256. Investigation of Fourier Series

257. Assumptions Regarding the Contour

258. Auxiliary Propositions

259. Transformation of the Contour Integrals

260. Proof of the Fundamental Lemma

261. Derivatives of Eigenfunctions

262. Proof of the Auxiliary Propositions

263. The Boundary Value Problem for a Sphere

264. Vibrations of the Interior Part of a Sphere

265. Investigation of the Solution

266. The Boundary Value Problem for the Telegraphist's Equation

Index

## Product details

- No. of pages: 826
- Language: English
- Copyright: © Pergamon 1964
- Published: January 1, 1964
- Imprint: Pergamon
- eBook ISBN: 9781483194714

## About the Author

### V. I. Smirnov

## About the Editor

### A. J. Lohwater

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