A Course of Higher Mathematics

A Course of Higher Mathematics

Adiwes International Series in Mathematics, Volume 4

1st Edition - January 1, 1964

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  • Author: V. I. Smirnov
  • eBook ISBN: 9781483194714

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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


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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