A Course of Higher Mathematics

IntroductionPreface to the Second EditionPreface to the Third EditionChapter 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 EquationsChapter 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. ExamplesChapter 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. ExamplesChapter 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 EquationIndex