Advanced calculus / by Ivan S. Sokolnikoff.
Editor: New York : McGraw-Hill, 1939Descripción: x, 446 p. ; cmOtra clasificación: 26-XXCONTENTS PreFacE v Chapter I Section LIMITS AND CONTINUITY 1. Number System [1] 2. Sequences [3] 3. Limit of a Sequence. Convergence [6] 4. Inequality of Bernoulli. lim n→∞ n√a = 1 [8] 5. Existence of the Limit. [10] 6. Criterion for Convergence of Monotone Sequences [14] 7. Divergent Sequences. Upper and Lower Limits [16] 8. Functions of a Single Variable [22] 9. Theorems on Limits [26] 10. The Base of the Natural Logarithms [28] 11. Continuity [31] 12. Properties of Continuous Functions [35] 13. Uniform Continuity [37] Chapter II DERIVATIVES AND DIFFERENTIALS 14. Derivatives [41] 15- Differentials [43] 16. Derivatives of Composite Functions [45] 17. Derivatives and Differentials of Higher Orders [47] 18. Fermat’s Theorem [49] 19. Rolle’s Theorem [50] 20. Mean-value Theorem [51] 21. Theorem of Cauchy. L’Hospital’s Rule [53] Chapter III FUNCTIONS OF SEVERAL VARIABLES 22. Limits and Continuity [58] 23. Partial Derivatives [62] 24. Differentiation of Composite Functions [67] 25. Differentiation of Composite and Implicit Functions [71] 26. Euler’s Theorem [75] 27. Directional Derivatives [76] 28. Tangent Plane and Normal Line to a Surface [80] 29. Space Curves [83] 30. Directional Derivatives in Space [86] 31. Partial Derivatives of Higher Order [87] 32. Higher Derivatives of Implicit Functions [89] 33. Change of Variables [91] Chapter IV DEFINITE INTEGRALS 34. Riemann Integral [99] 35. Riemann Integral—(Continued) [104] 36. Direct Evaluation of Integrals [110] 37. Mean-value Theorems for Integrals [113] 38. Fundamental Theorem of the Integral Calculus [118] 39. Differentiation under the Integral Sign [121] 40. Change of Variable [124] 41. Applications of Definite Integrals [126] Chapter V MULTIPLE INTEGRALS 42. Double Integrals [130] 43. Evaluation of the Double Integral [131] 44. Geometric Interpretation of the Double Integral [139] 45. Triple Integrals [142] 46. Change of Variables in a Double Integral [147] 47. Transformation of Points [153] 48. Change of Variables in a Triple Integral [155] 49. Spherical and Cylindrical Coordinates [157] 50. Surface Integrals [161] 51. Green’s Theorem in Space [167] 52. Symmetrical Form of Green’s Theorem [170] Chapter VI LINE INTEGRALS 53. Definition of Line Integral [174] 54. Area of a Closed Curve [178] 55. Green’s Theorem for the Plane [181] 56. Properties of Line Integrals [185] 57. Multiply Connected Regions [192] 58. Line Integrals in Space . [195] 59. Stokes’s Theorem [196] 60. Applications of Line Integrals [199] Chapter VII INFINITE SERIES 61. Infinite Series [209] 62. Series of Positive Terms [213] 63. More General Tests [225] 64. Series of Arbitrary Terms [233] 65. Absolute Convergence [236] 66. Properties of Absolutely Convergent Series [240] 67. Double Series [246] 68. Series of Functions. Uniform Convergence [247] 69. Geometric Interpretation of Uniform Convergence [252] 70. Properties of Uniformly Convergent Series [256] 71. Weierstrass Test for Uniform Convergence [262] 72. Abel’s Test for Uniform Convergence [264] Chapter VIII POWER SERIES 73. Power Series [267] 74. Interval of Convergence [269] 75- Properties of Functions Defined by Power Series [275] 76. Abel’s Theorem [276] 77. Uniqueness Theorem on Power Series [279] 78. Algebra of Power Series [280] 79. Calculations Involving Power Series [285] Chapter IX APPLICATIONS OF POWER SERIES 80. Extended Law of the Mean [291] 81. Taylor’s Formula [293] 82. Taylor’s Series [296] 83. Applications of Taylor’s Formula [298] 84. Euler’s Formulas and Hyperbolic Functions [306] 85. Integration of Power Series [309] 86. Evaluation of Definite Integrals [310] 87. Maxima and Minima of Functions of One Variable [315] 88. Taylor’s Formula for Functions of Several Variables [317] 89. Maxima and Minima, of Functions of Several Variables [321] 90. Constrained Maxima and Minima [327] 9)1. Lagrange’s Multipliers [331] Chapter X IMPROPER INTEGRALS 92. Integral with Infinite Limit [335] 93. Tests for Convergence of Integrals with Infinite Limits [341] 94. Integrals in Which the Integrand Becomes Infinite [347] 95. Tests for Convergence of Integrals Whose Integrands Become Infinite [350] 96. Operations with Improper Integrals [352] 97. Evaluation of Improper Integral's [357] 98. Improper Multiple Integrals [365] 99. Gamma Functions [372] Chapter XI FOURIER SERIES 100. Criterion of Approximation [378] 101. Fourier Coefficients [380] 102. Conditions of Dirichlet [385] 103. Orthogonal Functions [388] 104. Expansion of Functions in Fourier Series [390] 105. Sine and Cosine Series [397] 106. Extension of Interval of Expansion [401] 107. Complex Form of Fourier Series [403] 108. Differentiation and Integration of Fourier Series [405] 109. Fourier Integral [409] Chapter XII IMPLICIT FUNCTIONS 110. A Simple Problem in Implicit Functions [415] 111. Generalization of the Simple Problem [418] 112. Functional Dependence [423] 113. Existence Theorem for Implicit Functions [425] 114. Existence Theorem for Simultaneous Equations [430] 115. Functional Dependence [433] 116. Properties of Jacobians [438] Index [441]
Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
---|---|---|---|---|---|---|---|---|
Libros | Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 26 So683 (Browse shelf) | Available | A-139 |
CONTENTS --
PreFacE v --
Chapter I --
Section LIMITS AND CONTINUITY --
1. Number System [1] --
2. Sequences [3] --
3. Limit of a Sequence. Convergence [6] --
4. Inequality of Bernoulli. lim n→∞ n√a = 1 [8] --
5. Existence of the Limit. [10] --
6. Criterion for Convergence of Monotone Sequences [14] --
7. Divergent Sequences. Upper and Lower Limits [16] --
8. Functions of a Single Variable [22] --
9. Theorems on Limits [26] --
10. The Base of the Natural Logarithms [28] --
11. Continuity [31] --
12. Properties of Continuous Functions [35] --
13. Uniform Continuity [37] --
Chapter II --
DERIVATIVES AND DIFFERENTIALS --
14. Derivatives [41] --
15- Differentials [43] --
16. Derivatives of Composite Functions [45] --
17. Derivatives and Differentials of Higher Orders [47] --
18. Fermat’s Theorem [49] --
19. Rolle’s Theorem [50] --
20. Mean-value Theorem [51] --
21. Theorem of Cauchy. L’Hospital’s Rule [53] --
Chapter III --
FUNCTIONS OF SEVERAL VARIABLES --
22. Limits and Continuity [58] --
23. Partial Derivatives [62] --
24. Differentiation of Composite Functions [67] --
25. Differentiation of Composite and Implicit Functions [71] --
26. Euler’s Theorem [75] --
27. Directional Derivatives [76] --
28. Tangent Plane and Normal Line to a Surface [80] --
29. Space Curves [83] --
30. Directional Derivatives in Space [86] --
31. Partial Derivatives of Higher Order [87] --
32. Higher Derivatives of Implicit Functions [89] --
33. Change of Variables [91] --
Chapter IV --
DEFINITE INTEGRALS --
34. Riemann Integral [99] --
35. Riemann Integral—(Continued) [104] --
36. Direct Evaluation of Integrals [110] --
37. Mean-value Theorems for Integrals [113] --
38. Fundamental Theorem of the Integral Calculus [118] --
39. Differentiation under the Integral Sign [121] --
40. Change of Variable [124] --
41. Applications of Definite Integrals [126] --
Chapter V --
MULTIPLE INTEGRALS --
42. Double Integrals [130] --
43. Evaluation of the Double Integral [131] --
44. Geometric Interpretation of the Double Integral [139] --
45. Triple Integrals [142] --
46. Change of Variables in a Double Integral [147] --
47. Transformation of Points [153] --
48. Change of Variables in a Triple Integral [155] --
49. Spherical and Cylindrical Coordinates [157] --
50. Surface Integrals [161] --
51. Green’s Theorem in Space [167] --
52. Symmetrical Form of Green’s Theorem [170] --
Chapter VI --
LINE INTEGRALS --
53. Definition of Line Integral [174] --
54. Area of a Closed Curve [178] --
55. Green’s Theorem for the Plane [181] --
56. Properties of Line Integrals [185] --
57. Multiply Connected Regions [192] --
58. Line Integrals in Space . [195] --
59. Stokes’s Theorem [196] --
60. Applications of Line Integrals [199] --
Chapter VII --
INFINITE SERIES --
61. Infinite Series [209] --
62. Series of Positive Terms [213] --
63. More General Tests [225] --
64. Series of Arbitrary Terms [233] --
65. Absolute Convergence [236] --
66. Properties of Absolutely Convergent Series [240] --
--
67. Double Series [246] --
68. Series of Functions. Uniform Convergence [247] --
69. Geometric Interpretation of Uniform Convergence [252] --
70. Properties of Uniformly Convergent Series [256] --
71. Weierstrass Test for Uniform Convergence [262] --
72. Abel’s Test for Uniform Convergence [264] --
Chapter VIII --
POWER SERIES --
73. Power Series [267] --
74. Interval of Convergence [269] --
75- Properties of Functions Defined by Power Series [275] --
76. Abel’s Theorem [276] --
77. Uniqueness Theorem on Power Series [279] --
78. Algebra of Power Series [280] --
79. Calculations Involving Power Series [285] --
Chapter IX --
APPLICATIONS OF POWER SERIES --
80. Extended Law of the Mean [291] --
81. Taylor’s Formula [293] --
82. Taylor’s Series [296] --
83. Applications of Taylor’s Formula [298] --
84. Euler’s Formulas and Hyperbolic Functions [306] --
85. Integration of Power Series [309] --
86. Evaluation of Definite Integrals [310] --
87. Maxima and Minima of Functions of One Variable [315] --
88. Taylor’s Formula for Functions of Several Variables [317] --
89. Maxima and Minima, of Functions of Several Variables [321] --
90. Constrained Maxima and Minima [327] --
9)1. Lagrange’s Multipliers [331] --
Chapter X --
IMPROPER INTEGRALS --
92. Integral with Infinite Limit [335] --
93. Tests for Convergence of Integrals with Infinite Limits [341] --
94. Integrals in Which the Integrand Becomes Infinite [347] --
95. Tests for Convergence of Integrals Whose Integrands Become --
Infinite [350] --
96. Operations with Improper Integrals [352] --
97. Evaluation of Improper Integral's [357] --
98. Improper Multiple Integrals [365] --
99. Gamma Functions [372] --
Chapter XI --
FOURIER SERIES --
100. Criterion of Approximation [378] --
101. Fourier Coefficients [380] --
102. Conditions of Dirichlet [385] --
103. Orthogonal Functions [388] --
104. Expansion of Functions in Fourier Series [390] --
105. Sine and Cosine Series [397] --
106. Extension of Interval of Expansion [401] --
107. Complex Form of Fourier Series [403] --
108. Differentiation and Integration of Fourier Series [405] --
109. Fourier Integral [409] --
Chapter XII --
IMPLICIT FUNCTIONS --
110. A Simple Problem in Implicit Functions [415] --
111. Generalization of the Simple Problem [418] --
112. Functional Dependence [423] --
113. Existence Theorem for Implicit Functions [425] --
114. Existence Theorem for Simultaneous Equations [430] --
115. Functional Dependence [433] --
116. Properties of Jacobians [438] --
Index [441] --
MR, REVIEW #
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