A Course of Higher Mathematics

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.

Contenu

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

9. Adjoint Integral Equation

10. The Case of an Eigenvalue

    11· Fredholm Minors

11. Degenerate Equations

12. Examples

13. Generalization of the Results Obtained

14. The Selection Principle

15. The Selection Principle (Continued)

16. Unbounded Kernels

17. Integral Equations with Unbounded Kernels

18. The Case of an Eigenvalue

19. Equations with Continuous Iterated Kernels

20. Symmetric Kernels

21. Expansion in Eigenfunctions

22. Dini's Theorem

23. Expansion of Iterated Kernels

24. Classification of Symmetric Kernels

25. Extremal Properties of the Eigenfunctions

26. Mercer's Theorem

27. The Case of a Weakly Polar Kernel

28. Non-Homogeneous Equations

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

30. Hermitian Kernels

31. Equations Reducible to Symmetric Equations

32. Examples

33. Kernels Depending on a Parameter

34. Space of Continuous Functions

35. Linear Operators

36. Existence of the Eigenvalue

37. Sequences of Eigenvalues and Expansion Theorem

38. Space of Complex Continuous Functions

39. Completely Continuous Integral Operators

40. Normal Operators

41. The Case of Functions of Several Variables

42. Volterra's Equation

43. Laplace Transformation

44. Convolution of Functions

45. Volterra Equation of Special Type

46. Volterra Equation of the First Kind

47. Examples

48. Weighted Integral Equations

49. Integral Equation of the First Kind with Cauchy Kernels

50. Boundary Value Problems for Analytic Functions

51. Integral Equations of the Second Kind with Cauchy Kernels

52. Boundary Value Problems for the Case of a Segment

53. Inversion of a Cauchy Type Integral

54. Fourier's Integral Equation

55. Equations in the Case of an Infinite Interval

56. Examples

57. The Case of a Semi-Infinite Interval

58. Examples

59. 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

104. Characteristic Manifolds

105. Cauchy's Method

106. Cauchy's Problem

107. Uniqueness of the Solution

108. The Singular Case

109. Any Number of Independent Variables

110. Complete, General and Singular Integrals

111. the Complete Integral and Cauchy's Problem

112. Examples

113. The Case of Any Number of Variables

114. Jacobi's Theorem

115. Systems of Two First Order Equations

116. The Lagrange-Charpit Method

117. Systems of Linear Equations

118. Complete and Jacobian Systems

119. Integration of Complete Systems

120. Poisson Brackets

121. Jacobi's Method

122. Canonical Systems

123. Examples

124. The Method of Majorant Series

125. Kovalevskaya's Theorem

126. Equations of Higher Order

     § 2. Equations of Higher Orders

127. Types of Second Order Equation

128. Equations with Constant Coefficients

129. Normal Forms with Two Independent Variables

130. Cauchy's Problem

131. Characteristic Strips

132. Higher Order Derivatives

133. Real and Imaginary Characteristics

134. Fundamental Theorems

135. Intermediate Integrals

136. The Monge-Ampere Equations

137. Characteristics with Any Number of Independent Variables

138. Bicharacteristics

139. The Connection with Variational Problems

140. The Propagation of a Surface of Discontinuity

141. Strong Discontinuities

142. Riemann's Method

143. Characteristic Initial Data

144. Existence Theorems

145. Method of Successive Approximations

146. Green's Formula

147. Sobolev's Formula

148. Sobolev's Formula (Continued)

149. Construction of the Function A

150. The General Case of Initial Data

151. Generalized Wave Equation

152. The Case of Any Number of Independent Variables

153. Basic Inequalities

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

155. The Case of the Wave Equation

156. Supplementary Propositions

157. Generalized Solutions of the Wave Equation

158. Equations of the Elliptic Type

159. Generalized Solution of Poisson's Equation

     § 3. Systems of Equations

160. Characteristics of Systems of Equations

161. Kinematic Compatibility Conditions

162. Dynamic Compatibility Conditions

163. The Equations of Hydrodynamics

164. Equations of the Theory of Elasticity

165. Anisotropie Elastic Media

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