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 ¿ 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.