Mathematical analysis : a modern approach to advanced calculus / by Tom M. Apostol.
Series Addison-Wesley series in mathematicsEditor: Reading, Mass. : Addison-Wesley, c1957Descripción: xii, 559 p : il. ; 23 cmOtra clasificación: 26Bxx (26B05 26B10 26B15)CONTENTS Chapter 1. The Real and Complex Number Systems [1] 1—1 Introduction [1] 1-2 Arithmetical properties of real numbers [1] 1-3 Order properties of real numbers [2] 1—4 Geometrical representation of real numbers [2] 1-5 Decimal representation of real numbers [2] 1-6 Rational numbers [3] 1-7 Some irrational numbers [4] 1-8 Some fundamental inequalities [5] 1-9 Infimum and supremum [7] 1—10 Complex numbers [9] 1-11 Geometric representation of complex numbers [10] 1-12 The imaginary unit [11] 1-13 Absolute value of a complex number [12] 1—14 Impossibility of ordering the complex numbers [12] 1-15 Complex exponentials [13] 1—16 The argument of a complex number [14] 1—17 Integral powers and roots of complex numbers [15] 1-18 Complex logarithms [16] 1—19 Complex powers [17] 1- 20 Complex sines and cosines [18] Chapter 2. Some Basic Notions of Set Theory [24] 2- 1 Fundamentals of set theory [24] 2-2 Notations [24] 2-3 Ordered pairs [25] 2-4 Cartesian product of two sets [25] 2-5 Relations and functions in the plane [25] 2-6 General definition of relation [27] 2-7 General definition of function [27] 2-8 One-to-one functions and inverses [29] 2-9 Composite functions [30] 2-10 Sequences [30] 2-11 The number of elements in a set [31] 2- 12 Set algebra [33] Chapter 3. Elements of Point Set Theory [40] 3- 1 Introduction [40] 3-2 Intervals and open sets in E1 [40] 3-3 The structure of open sets in E1 [42] 3-4 Accumulation points and the Bolzano-Weierstrass theorem in E1 [43] 3-5 Closed sets in Ei [44] 3-6 Extensions to higher dimensions [45] 3-7 The Heine-Borel covering theorem [51] 3-8 Compactness [54] 3-9 Infinity in the real number system [56] 3- 10 Infinity in the complex plane [57] Chapter 4. The Limit Concept and Continuity [61] 4- 1 The definition of limit [61] 4-2 Some basic theorems on limits [64] 4-3 The Cauchy condition [65] 4-4 Algebra of limits [67] 4-5 Continuity [67] 4-6 Examples of continuous functions [69] 4-7 Functions continuous on open or closed sets [70] 4-8 Functions continuous on compact sets [71] 4-9 Topological mappings [72] 4-10 Properties of real-valued continuous functions [72] 4-11 Uniform continuity [74] 4-12 Discontinuities of real-valued functions [76] 4-13 Monotonic functions [77] 4- 14 Necessary and sufficient conditions for continuity [79] Chapter 5. Differentiation of Functions of One Real Variable [86] 5- 1 Introduction [86] 5-2 Definition of derivative [86] 5-3 Algebra of derivatives [88] 5-4 The chain rule [88] 5-5 One-sided derivatives and infinite derivatives [89] 5-6 Functions with nonzero derivative [91] 5-7 Functions with zero derivative [91] 5-8 Rolle’s theorem [92] 5-9 The Mean Value Theorem of differential calculus [93] 5—10 Intermediate value theorem for derivatives [94] 5- 11 Taylor’s formula with remainder [95] Chapter 6. Differentiation of Functions of Several Variables [103] 6- 1 Introduction [103] 6-2 The directional derivative [104] 6-3 Differentials of functions of one real variable [105] 6-4 Differentials of functions of several variables [107] 6-5 The gradient vector [110] 6-6 Differentials of composite functions and the chain rule [112] 6-7 Cauchy’s invariant rule [114] 6-8 The Mean Value Theorem for functions of several variables [117] 6-9 A sufficient condition for existence of the differential [118] 6-10 Partial derivatives of higher order [120] 6-11 Taylor’s formula for functions of several variables [123] 6-12 Differentiation of functions of a complex variable [125] 6- 13 The Cauchy-Riemann equations [126] Chapter 7. Applications of Partial Differentiation [138] 7- 1 Introduction [138] 7-2 Jacobians [139] 7-3 Functions with nonzero Jacobian [141] 7—4 The inverse function theorem [144] 7-5 The implicit function theorem [446] 7-6 Extremum problems [448] 7-7 Sufficient conditions for a local extremum [149] 7-8 Extremum problems with side conditions [452] Chapter 8. Functions of Bounded Variation, Rectifiable Curves and Connected Sets [162] 8-1 Introduction [462] 8-2 Properties of monotonic functions [162] 8-3 Functions of bounded variation [163] 8-4 Total variation [165] 8-5 Continuous functions of bounded variation [168] 8-6 Curves [169] 8-7 Equivalence of continuous vector-valued functions [170] 8-8 Directed paths [174] 8-9 Rectifiable curves [175] 8-10 Properties of arc length [176] 8-11 Connectedness [177] 8-12 Components of a set [182] 8-13 Regions [182] 8-14 Statement of the Jordan curve theorem and related results [183] Chapter 9. Theory of Riemann-Stieltjes Integration [191] 9-1 Introduction [191] 9-2 Notations [192] 9-3 The definition of the Riemann-Stieltjes integral [192] 9-4 Linearity properties [193] 9-5 Integration by parts [195] 9-6 Change of variable in a Riemann-Stieltjes integral [196] 9-7 Reduction to a Riemann integral [197] 9-8 Step functions as integrators [198] 9-9. Monotonically increasing integrators. Upper and lower integrals [202] 9-10 Riemann’s condition [206] 9-11 Integrators of bounded variation [207] 9-12 Sufficient conditions for existence of Riemann-Stieltjes integrals [211] 9-13 Necessary conditions for existence of Riemann-Stieltjes integrals [212] 9-14 Mean Value Theorems for Riemann-Stieltjes integrals [213] 9-15 The integral as a function of the interval [214] 9-16 Change of variable in a Riemann integral [215] 9-17 Second Mean Value Theorem for Riemann integrals [217] 9-18 Riemann-Stieltjes integrals depending on a parameter [217] 9-19 Differentiation under the integral sign [219] 9-20 Interchanging the order of integration [221] 9-21 Oscillation of a function [222] 9-22 Jordan content of bounded sets in [224] 9-23 A necessary and sufficient condition for integrability in terms of content [226] 9-24 Outer Lebesgue measure of subsets of [228] 9-25 A necessary and sufficient condition for integrability in terms of measure [230] 9-26 Complex-valued Riemann-Stieltjes integrals [231] 9-27 Contour integrals [232] 9-28 The winding number [236] 9-29 Orientation of rectifiable Jordan curves [240] Chapter 10. Multiple Integrals and Line Integrals [251] 10-1 Introduction [251] 10-2 The measure (or content) of elementary sets in En [251] 10-3 Riemann integration of bounded functions defined on intervals in En [252] 10-4 Jordan content of bounded sets in En [255] 10-5 Necessary and sufficient conditions for the existence of multiple integrals [258] 10-6 Evaluation of a multiple integral by repeated integration [260] 10-7 Multiple integration over more general sets [266] 10-8 Mean Value Theorem for multiple integrals [269] 10-9 Change of variable in a multiple integral [270] 10-10 Line integrals [275] . 10-11 Line integrals with respect to arc length [279] 10-12 The line integral of a gradient [279] 10-13 Green’s theorem for rectangles [283] 10-14 Green’s theorem for regions bounded by rectifiable Jordan curves [284] 10- 15 Independence of the path [292] Chapter 11. Vector Analysis [304] 11- 1 Introduction [304] 11-2 Linear independence and bases in En [304] 11-3 Geometric representation of vectors in E3 [306] 11-4 Geometric interpretation of the dot product in E3 [308] 11-5 The cross product of vectors in A3 [308] 11-6 The scalar triple product [309] 11-7 Derivatives of vector-valued functions [314] 11-8 Elementary differential geometry of space curves [314] 11-9 The tangent vector of a curve [315] 11-10 Normal vectors, curvature, torsion [316] 11-11 Vector fields [318] 11-12 The gradient field in [319] 11—13 The curl of a vector field in E3 [320] 11—14 The divergence of a vector field in En [322] 11-15 The Laplacian operator [324] 11-16 Surfaces [325] 11-17 Explicit representation of a parametric surface [329] 11-18 Area of a parametric surface [330] 11-19 The sum of parametric surfaces [331] 11-20 Surface integrals [332] 11-21 The theorem of Stokes [334] 11-22 Orientation of surfaces [338] 11-23 Gauss’ theorem (the divergence theorem) [339] 11-24 Coordinate transformations [341] Chapter 12. Infinite Series and Infinite Products [353] 12-1 Introduction [353] 12-2 Convergent and divergent sequences [353] 12-3 Limit superior and limit inferior of a real-valued sequence [353] 12-4 Monotonic sequences of real numbers [355] 12-5 Infinite series [355] 12-6 Inserting and removing parentheses [357] 12-7 Alternating series [358] 12-8 Absolute and conditional convergence [359] 12-9 Real and imaginary parts of a complex series [360] 12-10 Tests for convergence of series with positive terms [360] 12-11 The ratio test and the root test [363] 12-12 Dirichlet’s test and Abel’s test [364] 12-13 Rearrangements of series [367] 12-14 Double sequences [371] 12-15 Double series [372] 12-16 Multiplication of series [376] 12-17 Cesàro summability [378] 12- 18 Infinite products [379] Chapter 13. Sequences of Functions [390] 13- 1 Introduction [390] 13-2 Examples of sequences of real-valued functions [391] 13-3 Definition of uniform convergence [392] 13-4 An application to double sequences [394] 13-5 Uniform convergence and continuity [394] 13-6 The Cauchy condition for uniform convergence [395] $3^-7 Uniform convergence of infinite series [395] 13-8 A space-filling curve [396] 13-9 An application to repeated series [398] 13-10 Uniform convergence and Riemann-Stieltjes integration [399] 13-11 Uniform convergence and differentiation [401] 13-12 Sufficient conditions for uniform convergence of a series [403] 13-13 Bounded convergence. Arzelà’s theorem [405] 13-14 Mean convergence [407] 13-15 Power series [409] 13-16 Multiplication of .power series [413] 13-17 The substitution theorem [414] 13-18 Real power series [416] 13-1'9 Bernstein’s theorem [418] 13-2.0 The binomial series [420] 13-21 Abel’s limit theorem [421] 13- 22 Tauber’s theorem [423] Chapter 14. Improper Riemann-Stieltjes Integrals [429] 14- 1 Introduction [429] 14-2 Infinite Riemann-Stieltjes integrals [429] 14-3 Tests for convergence of infinite integrals [431] 14-4 Infinite series and infinite integrals [434] 14-5 Improper integrals of the second kind [435]
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Referencias bibliográficas al final de cada capítulo.
CONTENTS --
Chapter 1. The Real and Complex Number Systems [1] --
1—1 Introduction [1] --
1-2 Arithmetical properties of real numbers [1] --
1-3 Order properties of real numbers [2] --
1—4 Geometrical representation of real numbers [2] --
1-5 Decimal representation of real numbers [2] --
1-6 Rational numbers [3] --
1-7 Some irrational numbers [4] --
1-8 Some fundamental inequalities [5] --
1-9 Infimum and supremum [7] --
1—10 Complex numbers [9] --
1-11 Geometric representation of complex numbers [10] --
1-12 The imaginary unit [11] --
1-13 Absolute value of a complex number [12] --
1—14 Impossibility of ordering the complex numbers [12] --
1-15 Complex exponentials [13] --
1—16 The argument of a complex number [14] --
1—17 Integral powers and roots of complex numbers [15] --
1-18 Complex logarithms [16] --
1—19 Complex powers [17] --
1- 20 Complex sines and cosines [18] --
Chapter 2. Some Basic Notions of Set Theory [24] --
2- 1 Fundamentals of set theory [24] --
2-2 Notations [24] --
2-3 Ordered pairs [25] --
2-4 Cartesian product of two sets [25] --
2-5 Relations and functions in the plane [25] --
2-6 General definition of relation [27] --
2-7 General definition of function [27] --
2-8 One-to-one functions and inverses [29] --
2-9 Composite functions [30] --
2-10 Sequences [30] --
2-11 The number of elements in a set [31] --
2- 12 Set algebra [33] --
Chapter 3. Elements of Point Set Theory [40] --
3- 1 Introduction [40] --
3-2 Intervals and open sets in E1 [40] --
3-3 The structure of open sets in E1 [42] --
3-4 Accumulation points and the Bolzano-Weierstrass theorem in E1 [43] --
3-5 Closed sets in Ei [44] --
3-6 Extensions to higher dimensions [45] --
3-7 The Heine-Borel covering theorem [51] --
3-8 Compactness [54] --
3-9 Infinity in the real number system [56] --
3- 10 Infinity in the complex plane [57] --
Chapter 4. The Limit Concept and Continuity [61] --
4- 1 The definition of limit [61] --
4-2 Some basic theorems on limits [64] --
4-3 The Cauchy condition [65] --
4-4 Algebra of limits [67] --
4-5 Continuity [67] --
4-6 Examples of continuous functions [69] --
4-7 Functions continuous on open or closed sets [70] --
4-8 Functions continuous on compact sets [71] --
4-9 Topological mappings [72] --
4-10 Properties of real-valued continuous functions [72] --
4-11 Uniform continuity [74] --
4-12 Discontinuities of real-valued functions [76] --
4-13 Monotonic functions [77] --
4- 14 Necessary and sufficient conditions for continuity [79] --
Chapter 5. Differentiation of Functions of One Real Variable [86] --
5- 1 Introduction [86] --
5-2 Definition of derivative [86] --
5-3 Algebra of derivatives [88] --
5-4 The chain rule [88] --
5-5 One-sided derivatives and infinite derivatives [89] --
5-6 Functions with nonzero derivative [91] --
5-7 Functions with zero derivative [91] --
5-8 Rolle’s theorem [92] --
5-9 The Mean Value Theorem of differential calculus [93] --
5—10 Intermediate value theorem for derivatives [94] --
5- 11 Taylor’s formula with remainder [95] --
Chapter 6. Differentiation of Functions of Several Variables [103] --
6- 1 Introduction [103] --
6-2 The directional derivative [104] --
6-3 Differentials of functions of one real variable [105] --
6-4 Differentials of functions of several variables [107] --
6-5 The gradient vector [110] --
6-6 Differentials of composite functions and the chain rule [112] --
6-7 Cauchy’s invariant rule [114] --
6-8 The Mean Value Theorem for functions of several variables [117] --
6-9 A sufficient condition for existence of the differential [118] --
6-10 Partial derivatives of higher order [120] --
6-11 Taylor’s formula for functions of several variables [123] --
6-12 Differentiation of functions of a complex variable [125] --
6- 13 The Cauchy-Riemann equations [126] --
Chapter 7. Applications of Partial Differentiation [138] --
7- 1 Introduction [138] --
7-2 Jacobians [139] --
7-3 Functions with nonzero Jacobian [141] --
7—4 The inverse function theorem [144] --
7-5 The implicit function theorem [446] --
7-6 Extremum problems [448] --
7-7 Sufficient conditions for a local extremum [149] --
7-8 Extremum problems with side conditions [452] --
Chapter 8. Functions of Bounded Variation, Rectifiable Curves and Connected Sets [162] --
8-1 Introduction [462] --
8-2 Properties of monotonic functions [162] --
8-3 Functions of bounded variation [163] --
8-4 Total variation [165] --
8-5 Continuous functions of bounded variation [168] --
8-6 Curves [169] --
8-7 Equivalence of continuous vector-valued functions [170] --
8-8 Directed paths [174] --
8-9 Rectifiable curves [175] --
8-10 Properties of arc length [176] --
8-11 Connectedness [177] --
8-12 Components of a set [182] --
8-13 Regions [182] --
8-14 Statement of the Jordan curve theorem and related results [183] --
Chapter 9. Theory of Riemann-Stieltjes Integration [191] --
9-1 Introduction [191] --
9-2 Notations [192] --
9-3 The definition of the Riemann-Stieltjes integral [192] --
9-4 Linearity properties [193] --
9-5 Integration by parts [195] --
9-6 Change of variable in a Riemann-Stieltjes integral [196] --
9-7 Reduction to a Riemann integral [197] --
9-8 Step functions as integrators [198] --
9-9. Monotonically increasing integrators. Upper and lower integrals [202] --
9-10 Riemann’s condition [206] --
9-11 Integrators of bounded variation [207] --
9-12 Sufficient conditions for existence of Riemann-Stieltjes integrals [211] --
9-13 Necessary conditions for existence of Riemann-Stieltjes integrals [212] --
9-14 Mean Value Theorems for Riemann-Stieltjes integrals [213] --
9-15 The integral as a function of the interval [214] --
9-16 Change of variable in a Riemann integral [215] --
9-17 Second Mean Value Theorem for Riemann integrals [217] --
9-18 Riemann-Stieltjes integrals depending on a parameter [217] --
9-19 Differentiation under the integral sign [219] --
9-20 Interchanging the order of integration [221] --
9-21 Oscillation of a function [222] --
9-22 Jordan content of bounded sets in [224] --
9-23 A necessary and sufficient condition for integrability in terms of content [226] --
9-24 Outer Lebesgue measure of subsets of [228] --
9-25 A necessary and sufficient condition for integrability in terms of measure [230] --
9-26 Complex-valued Riemann-Stieltjes integrals [231] --
9-27 Contour integrals [232] --
9-28 The winding number [236] --
9-29 Orientation of rectifiable Jordan curves [240] --
Chapter 10. Multiple Integrals and Line Integrals [251] --
10-1 Introduction [251] --
10-2 The measure (or content) of elementary sets in En [251] --
10-3 Riemann integration of bounded functions defined on intervals in En [252] --
10-4 Jordan content of bounded sets in En [255] --
10-5 Necessary and sufficient conditions for the existence of multiple integrals [258] --
10-6 Evaluation of a multiple integral by repeated integration [260] --
10-7 Multiple integration over more general sets [266] --
10-8 Mean Value Theorem for multiple integrals [269] --
10-9 Change of variable in a multiple integral [270] --
10-10 Line integrals [275] --
. 10-11 Line integrals with respect to arc length [279] --
10-12 The line integral of a gradient [279] --
10-13 Green’s theorem for rectangles [283] --
10-14 Green’s theorem for regions bounded by rectifiable Jordan curves [284] --
10- 15 Independence of the path [292] --
Chapter 11. Vector Analysis [304] --
11- 1 Introduction [304] --
11-2 Linear independence and bases in En [304] --
11-3 Geometric representation of vectors in E3 [306] --
11-4 Geometric interpretation of the dot product in E3 [308] --
11-5 The cross product of vectors in A3 [308] --
11-6 The scalar triple product [309] --
11-7 Derivatives of vector-valued functions [314] --
11-8 Elementary differential geometry of space curves [314] --
11-9 The tangent vector of a curve [315] --
11-10 Normal vectors, curvature, torsion [316] --
11-11 Vector fields [318] --
11-12 The gradient field in [319] --
11—13 The curl of a vector field in E3 [320] --
11—14 The divergence of a vector field in En [322] --
11-15 The Laplacian operator [324] --
11-16 Surfaces [325] --
11-17 Explicit representation of a parametric surface [329] --
11-18 Area of a parametric surface [330] --
11-19 The sum of parametric surfaces [331] --
11-20 Surface integrals [332] --
11-21 The theorem of Stokes [334] --
11-22 Orientation of surfaces [338] --
11-23 Gauss’ theorem (the divergence theorem) [339] --
11-24 Coordinate transformations [341] --
Chapter 12. Infinite Series and Infinite Products [353] --
12-1 Introduction [353] --
12-2 Convergent and divergent sequences [353] --
12-3 Limit superior and limit inferior of a real-valued sequence [353] --
12-4 Monotonic sequences of real numbers [355] --
12-5 Infinite series [355] --
12-6 Inserting and removing parentheses [357] --
12-7 Alternating series [358] --
12-8 Absolute and conditional convergence [359] --
12-9 Real and imaginary parts of a complex series [360] --
12-10 Tests for convergence of series with positive terms [360] --
12-11 The ratio test and the root test [363] --
12-12 Dirichlet’s test and Abel’s test [364] --
12-13 Rearrangements of series [367] --
12-14 Double sequences [371] --
12-15 Double series [372] --
12-16 Multiplication of series [376] --
12-17 Cesàro summability [378] --
12- 18 Infinite products [379] --
Chapter 13. Sequences of Functions [390] --
13- 1 Introduction [390] --
13-2 Examples of sequences of real-valued functions [391] --
13-3 Definition of uniform convergence [392] --
13-4 An application to double sequences [394] --
13-5 Uniform convergence and continuity [394] --
13-6 The Cauchy condition for uniform convergence [395] --
$3^-7 Uniform convergence of infinite series [395] --
13-8 A space-filling curve [396] --
13-9 An application to repeated series [398] --
13-10 Uniform convergence and Riemann-Stieltjes integration [399] --
13-11 Uniform convergence and differentiation [401] --
13-12 Sufficient conditions for uniform convergence of a series [403] --
13-13 Bounded convergence. Arzelà’s theorem [405] --
13-14 Mean convergence [407] --
13-15 Power series [409] --
13-16 Multiplication of .power series [413] --
13-17 The substitution theorem [414] --
13-18 Real power series [416] --
13-1'9 Bernstein’s theorem [418] --
13-2.0 The binomial series [420] --
13-21 Abel’s limit theorem [421] --
13- 22 Tauber’s theorem [423] --
Chapter 14. Improper Riemann-Stieltjes Integrals [429] --
14- 1 Introduction [429] --
14-2 Infinite Riemann-Stieltjes integrals [429] --
14-3 Tests for convergence of infinite integrals [431] --
14-4 Infinite series and infinite integrals [434] --
14-5 Improper integrals of the second kind [435] --
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