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A Course in Mathematical Analysis Volume 3 Variation of Solutions; Partial Differential Equations of the Second Order; Integral Equations; Calculus of Variations book cover, ISBN 9780486446523

A Course in Mathematical Analysis Volume 3 Variation of Solutions; Partial Differential Equations of the Second Order; Integral Equations; Calculus of Variations

by Goursat, Edouard; Bergmann, Howard G.

ISBN13: 9780486446523
ISBN10: 0486446522
Format: Hardcover
Copyright: 2013-04-04
Publisher: Dover Publications

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1. A Course in Mathematical Analysis Volume 3 Variation of Solutions; Partial Differential Equations of the Second Order; Integral Equations; Calculus of Variations > ISBN13: 9780486446523

List Price: $85.00

Summary

Eacute;douard Goursat's three-volumeA Course in Mathematical Analysisremains a classic study and a thorough treatment of the fundamentals of calculus. As an advanced text for students with one year of calculus, it offers an exceptionally lucid exposition. Volume 1 covers applications to geometry, expansion in series, definite integrals, and derivatives and differentials. Volume 2 explores functions of a complex variable and differential equations. Volume 3 surveys variations of solutions and partial differential equations of the second order and integral equations and calculus of variations. All volumes are 55/8 x 81/2, hardbound editions. Volume 1: 1904 ed. Index. 52 figures. 560pp. 0-486-44650-6 Volume 2: 1916 and 1917 eds. Index. 39 figures. 576pp. 0-486-44651-4 Volume 3: 1956 ed. 28 figures. 752pp. 0-486-44652-2

Table of Contents

CHAPTER I. VARIATION OF SOLUTIONS

1

(46)

I. EQUATIONS OF VARIATION

2

(22)

1. Comments on linear equations

2

(2)

2. Application to a semi-linear system

4

(3)

3. Integrals considered as functions of the initial values

7

(5)

4. Extension to equations which depend on parameters

12

(1)

5. Variations of solutions

13

(5)

6. Equations of variation

18

(2)

7. Poincaré's theorem

20

(4)

II. PERIODIC AND ASYMPTOTIC SOLUTIONS. STABILITY

24

(23)

8. Periodic solutions

24

(4)

9. Stable and unstable solutions

28

(3)

10. General theorems on stability

31

(3)

11. Application of general theorems

34

(5)

12. Stability of equilibrium

39

(2)

13. Applications to more general systems

41

(1)

14. Asymptotic series. Conditional stability

42

(3)

Comments and exercises

45

(2)

CHAPTER II. EQUATIONS OF MONGE–AMPÈRE

47

(42)

I. CHARACTERISTICS. INTERMEDIATE INTEGRALS.

47

(25)

15. Cauchy's problem for an equation of the second order

47

(6)

16. Surface elements. Manifold M.

53

(1)

17. Equations of Monge–Ampère. Characteristics

54

(5)

18. Properties of characteristics

59

(2)

19. Intermediate integrals

61

(6)

20. Miscellaneous applications. Examples

67

(5)

II. LAPLACE'S METHOD. CLASSIFICATION OF LINEAR EQUATIONS

72

(17)

21. Intermediate integrals of a linear equation

72

(3)

22. Laplace transformations

75

(4)

23. The three types of linear equations

79

(5)

24. Study of Cauchy's problem in a special case

84

(3)

Exercises

87

(2)

CHAPTER III. LINEAR EQUATIONS IN n VARIABLES

89

(24)

I. CLASSIFICATION OF EQUATIONS IN n VARIABLES

89

(9)

25. Characteristics of equations in n variables

89

(3)

26. Wave propagation

92

(3)

27. Generalities on completely linear equations

95

(3)

II. APPLICATIONS TO A FEW EXAMPLES

98

(15)

28. The sound equation

98

(5)

29. Cylindrical waves

103

(2)

30. Propagation of heat in an unbounded medium

105

(3)

31. The problem of the loop

108

(2)

32. Cooling of a sphere

110

(2)

Comments and exercises

112

(1)

CHAPTER IV. LINEAR EQUATIONS OF THE HYPERBOLIC TYPE

113

(57)

I STUDY OF SOME PROBLEMS RELATING TO THE EQUATION 8 = ƒ(x,y)

113

(20)

33. Determination of an integral from Cauchy's data

113

(6)

34. Mixed problems

119

(3)

35. Determination of an integral from its values along

122

(2)

36. Rectilinear motion of a gas

124

(5)

37. Vibrating strings

129

(4)

II. SUCCESSIVE APPROXIMATIONS. RIEMANN'S METHOD

133

(27)

38. Determination of an integral from its values along two characteristics

133

(4)

39. Riemann's function

137

(5)

40. First solution of Cauchy's problem

142

(4)

41. The adjoint equation

146

(2)

42. Riemann's method

148

(5)

43. Equations with constant coefficients

153

(4)

44. Other problems

157

(3)

III. EQUATIONS IN MORE THAN TWO VARIABLES

160

(10)

45. Fundamental formula

160

(2)

46. Volterra's method

162

(6)

Comments and exercises

168

(2)

CHAPTER V. LINEAR EQUATIONS OF ELLIPTIC TYPE

170

(74)

I. HARMONIC FUNCTIONS. POISSON'S INTEGRAL

170

(29)

47. General properties

170

(6)

48. Uniformly convergent integrals

176

(2)

49. Logarithmic potential

178

(3)

50. Green's second formula

181

(2)

51. Applications to harmonic functions

183

(3)

52. Poisson's integral

186

(4)

53. Relations to Fourier series

190

(2)

54. Harnack's theorem

192

(3)

55. Analytic continuation of a harmonic function

195

(4)

II. DIRICHLET'S PROBLEM. GREEN'S FUNCTION

199

(27)

56. Riemann's proof

199

(2)

57. C. Neumann's method

201

(5)

58. Generalization of the problem

206

(4)

59. Schwarz's alternate method

210

(4)

60. Exterior problem

214

(2)

61. Conformal representation

216

(3)

62. Green's function

219

(4)

63. Properties of Green's function

223

(3)

III. GENERAL EQUATION OF THE ELLIPTIC TYPE

226

(18)

64. Extension of Dirichlet's problem

226

(2)

65. Study of the equation Δu = ƒ(x,y)

228

(2)

66. Picard's method

230

(2)

67. Green's function for the general equation of elliptic type

232

(3)

68. Positive mixed problems

235

(2)

Comments and exercises

237

(7)

CHAPTER VI. HARMONIC FUNCTIONS IN THREE VARIABLES

244

(46)

I. DIRICHLET'S PROBLEM IN SPACE

244

(32)

69. General properties

244

(2)

70. Newtonian potential of single layer

246

(4)

71. Potential of double layer

250

(3)

72. Green's second formula

253

(4)

73. The interior problem and the exterior problem

257

(3)

74. Solution of the problem for the sphere

260

(2)

75. Laplace functions

262

(4)

76. Properties of the functions Yn

266

(3)

77. C. Neumann's method

269

(3)

78. Green's function

272

(4)

II. NEWTONIAN POTENTIAL

276

(14)

79. Potential of volume

276

(4)

80. Poisson's formula

280

(3)

81. Gauss's formula

283

(1)

82. Normal derivatives of a potential of single layer

284

(3)

83. Newtonian potential of double layer

287

(1)

Comments and exercises

288

(2)

CHAPTER VII. THE HEAT EQUATION

290

(35)

84. General remarks. Particular integrals

290

(3)

85. Analytic integrals

293

(3)

86. Fundamental solution

296

(2)

87. Poisson's formula

298

(6)

88. Integrals analogous to the potential

304

(6)

89. Extension of Green's formula. Applications

310

(5)

90. Properties of integrals

315

(3)

91. Problems with boundary conditions

318

(3)

Comments and exercises

321

(4)

Index

325

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