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9780486446523

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

by ;
  • ISBN13:

    9780486446523

  • ISBN10:

    0486446522

  • Format: Hardcover
  • Copyright: 2013-04-04
  • Publisher: Dover Publications
  • Purchase Benefits
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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