Mathematical analysis : a modern approach to advanced calculus / by Tom M. Apostol.

Por: Apostol, Tom MSeries Addison-Wesley series in mathematicsEditor: Reading, Mass. : Addison-Wesley, c1957Descripción: xii, 559 p : il. ; 23 cmOtra clasificación: 26Bxx (26B05 26B10 26B15)
Contenidos:
 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]
    Average rating: 0.0 (0 votes)
Item type Home library Shelving location Call number Materials specified Copy number Status Date due Barcode Course reserves
Libros Libros Instituto de Matemática, CONICET-UNS
Libros ordenados por tema 26 Ap645 (Browse shelf) Available A-6051

ANÁLISIS MATEMÁTICO III

Libros Libros Instituto de Matemática, CONICET-UNS
26 Ap645 (Browse shelf) Ej. 2 Available A-8053

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] --

MR, REVIEW #

There are no comments on this title.

to post a comment.

Click on an image to view it in the image viewer

¿Necesita ayuda?

Si necesita ayuda para encontrar información, puede visitar personalmente la biblioteca en Av. Alem 1253 Bahía Blanca, llamarnos por teléfono al 291 459 5116, o enviarnos un mensaje a biblioteca.antonio.monteiro@gmail.com

Powered by Koha