Advanced calculus / by Ivan S. Sokolnikoff.

Por: Sokolnikoff, Ivan Stephen, 1901-Editor: New York : McGraw-Hill, 1939Descripción: x, 446 p. ; cmOtra clasificación: 26-XX
Contenidos:
 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]
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ANÁLISIS MATEMÁTICO III

MATEMÁTICA ESPECIAL I


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] --

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