Advanced calculus / Angus E. Taylor, W. Robert Mann.

Por: Taylor, Angus E. (Angus Ellis), 1911-1999Colaborador(es): Mann, W. Robert (William Robert), 1920-Editor: New York : Wiley, c1983Edición: 3rd edDescripción: xv, 732 p. : il. ; 24 cmISBN: 0471025666Tema(s): CalculusOtra clasificación: 26-01 Recursos en línea: Publisher description | Table of Contents
Contenidos:
 CONTENTS
1 FUNDAMENTALS OF ELEMENTARY CALCULUS
1. Introduction [1]
1.1 Functions [2]
1.11 Derivatives [12]
1.12 Maxima and Minima [20]
1.2 The Law of the Mean (The Mean-Value Theorem for Derivatives) [26]
1.3 Differentials [32]
1.4 The Inverse of Differentiation [35]
1.5 Definite Integrals [38]
1.51 The Mean-Value Theorem for Integrals [45]
1.52 Variable Limits of Integration [46]
1.53 The Integral of a Derivative [49]
1.6 Limits [53]
1.61 Limits of Functions of a Continuous Variable [54]
1.62 Limits of Sequences [58]
1.63 The Limit Defining a Definite Integral [67]
1.64 The Theorem on Limits of Sums, Products, and Quotients [67]
2 THE REAL NUMBER SYSTEM
2. Numbers [72]
2.1 The Field of Real Numbers [72]
2.2 Inequalities. Absolute Value [74]
2.3 The Principle of Mathematical Induction [75]
2.4 The Axiom of Continuity [77]
2.5 Rational and Irrational Numbers [78]
2.6 The Axis of Reals [79]
2.7 Least Upper Bounds [80]
2.8 Nested Intervals [82]
3 CONTINUOUS FUNCTIONS
3. Continuity [85]
3.1 Bounded Functions [86]
3.2 The Attainment of Extreme Values [88]
3.3 The Intermediate-Value Theorem [90]
4 EXTENSIONS OF THE LAW OF THE MEAN
4. Introduction [95]
4.1 Cauchy’s Generalized Law of the Mean [95]
4.2 Taylor’s Formula with Integral Remainder [97]
4.3 Other Forms of the Remainder [99]
4.4 An Extension of the Mean-Value Theorem for Integrals [105]
4.5 L’Hospital’s Rule [106]
5 FUNCTIONS OF SEVERAL VARIABLES
5. Functions and Their Regions of Definition [116]
5.1 Point Sets [117]
5.2 Limits [122]
5.3 Continuity [125]
5.4 Modes of Representing a Function [127]
6 THE ELEMENTS OF PARTIAL DIFFERENTIATION
6. Partial Derivatives [130]
6.1 Implicit Functions [132]
6.2 Geometrical Significance of Partial Derivatives [135]
6.3 Maxima and Minima [138]
6.4 Differentials [144]
6.5 Composite Functions and the Chain Rule [154]
6.51 An Application in Fluid Mechanics [162]
6.52 Second Derivatives by the Chain Rule [164]
6.53 Homogeneous Functions. Euler’s Theorem [168]
6.6 Derivatives of Implicit Functions [172]
6.7 Extremal Problems with Constraints [177]
6.8 Lagrange’s Method [182]
6.9 Quadratic Forms [189]
7 GENERAL THEOREMS OF PARTIAL DIFFERENTIATION
7. Preliminary Remarks [196]
7.1 Sufficient Conditions for Differentiability [197]
7.2 Changing the Order of Differentiation [199]
7.3 Differentials of Composite Functions [202]
7.4 The Law of the Mean [204]
7.5 Taylor’s Formula and Series [207]
7.6 Sufficient Conditions for a Relative Extreme [211]
8 IMPLICIT-FUNCTION THEOREMS
8. The Nature of the Problem of Implicit Functions [222]
8.1 The Fundamental Theorem [224]
8.2 Generalization of the Fundamental Theorem [227]
8.3 Simultaneous Equations [230]
9 THE INVERSE FUNCTION THEOREM WITH APPLICATIONS
9.Introduction [237]
9.1 The Inverse Function Theorem in Two Dimensions [241]
9.2 Mappings [247]
9.3 Successive Mappings [252]
9.4 Transformations of Co-ordinates [255]
9.5 Curvilinear Co-ordinates [258]
9.6 Identical Vanishing of the Jacobian. Functional Dependence [263]
10 VECTORS AND VECTOR FIELDS
10. Purpose of the Chapter [268]
10.1 Vectors in Euclidean Space [268]
10.11 Orthogonal Unit Vectors in R3 [273]
10.12 The Vector Space Rn [274]
10.2 Cross Products in R3 [280]
10.3 Rigid Motions of the Axes [283]
10.4 Invariants [286]
10.5 Scalar Point Functions [291]
10.51 Vector Point Functions [293]
10.6 The Gradient of a Scalar Field [295]
10.7 The Divergence of a Vector Field [300]
10.8 The Curl of a Vector Field [305]
11 I LINEAR TRANSFORMATIONS
11. Introduction [309]
11.1 Linear Transformations [312]
11.2 The Vector Space L(Rn, Rm) [313]
11.3 Matrices and Linear Transformations [313]
11.4 Some Special Cases [316]
11.5 Norms [318]
11.6 Metrics [319]
11.7 Open Sets and Continuity [320]
11.8 A Norm on L(Rn,Rm) [324]
11.9 L(Rn) [327]
11.10 The Set of Invertible Operators [330]
12 DIFFERENTIAL CALCULUS OF FUNCTIONS FROM Rn TO Rm
12. Introduction [335]
12.1 The Differential and the Derivative [336]
12.2 The Component Functions and Differentiability [340]
12.21 Directional Derivatives and the Method of Steepest Descent [343]
12.3 Newton’s Method [347]
12.4 A Form of the Law of the Mean for Vector Functions [350]
12.41 The Hessian and Extreme Values [352]
12.5 Continuously Differentiable Functions [354]
12.6 The Fundamental Inversion Theorem [355]
12.7 The Implicit Function Theorem [361]
12.8 Differentiation of Scalar Products of Vector Valued Functions of a Vector Variable [366]
13 DOUBLE AND TRIPLE INTEGRALS
13. Preliminary Remarks [376]
13.1 Motivations [376]
13.2 Definition of a Double Integral [379]
13.21 Some Properties of the Double Integral [381]
13.22 Inequalities. The Mean-Value Theorem [382]
13.23 A Fundamental Theorem [383]
13.3 Iterated Integrals. Centroids [384]
13.4 Use of Polar Co-ordinates [390]
13.5 Applications of Double Integrals [395]
13.51 Potentials and Force Fields [401]
13.6 Triple Integrals [404]
13.7 Applications of Triple Integrals [409]
13.8 Cylindrical Co-ordinates [412]
13.9 Spherical Co-ordinates [413]
14 CURVES AND SURFACES
14. Introduction [417]
14.1 Representations of Curves [417]
14.2 Arc Length [418]
14.3 The Tangent Vector [421]
14.31 Principal normal. Curvature [423]
14.32 Binormal. Torsion [425]
14.4 Surfaces [428]
14.5 Curves on a Surface [433]
14.6 Surface Area [437]
15 LINE AND SURFACE INTEGRALS
15. Introduction [445]
15.1 Point Functions on Curves and Surfaces [445]
15.12 Line Integrals [446]
15.13 Vector Functions and Line Integrals. Work [451]
15.2 Partial Derivatives at the Boundary of a Region [455]
15.3 Green’s Theorem in the Plane [457]
15.31 Comments on the Proof of Green’s Theorem [463]
15.32 Transformations of Double Integrals [465]
15.4 Exact Differentials [469]
15.41 Line Integrals Independent of the Path [474]
15.5 Further Discussion of Surface Area [478]
15.51 Surface Integrals [480]
15.6 The Divergence Theorem [484]
15.61 Green’s Identities [492]
15.62 Transformation of Triple Integrals [494]
15.7 Stokes’s Theorem [499]
15.8 Exact Differentials in Three Variables [505]
16 POINT-SET THEORY
16. Preliminary Remarks [512]
16.1 Finite and Infinite Sets [512]
16.2 Point Sets on a Line [514]
16.3 The Bolzano-Weierstrass Theorem [517]
16.31 Convergent Sequences on a Line [518]
16.4 Point Sets in Higher Dimensions [520]
16.41 Convergent Sequences in Higher Dimensions [521]
16.5 Cauchy’s Convergence Condition [522]
16.6 The Heine-Borel Theorem [523]
17 FUNDAMENTAL THEOREMS ON CONTINUOUS FUNCTIONS
17. Purpose of the Chapter [527]
17.1 Continuity and Sequential Limits [527]
17.2 The Boundedness Theorem [529]
17.3 The Extreme-Value Theorem [529]
17.4 Uniform Continuity [529]
17.5 Continuity of Sums, Products, and Quotients [532]
17.6 Persistence of Sign [532]
17.7 The In termediate-Value Theorem [533]
18 THE THEORY OF INTEGRATION
18. The Nature of the Chapter [535]
18.1 The Definition of Integrability [535]
18.11 The Integrability of Continuous Functions [539]
18.12 Integrable Functions with Discontinuities [540]
18.2 The Integral as a Limit of Sums [542]
18.21 Duhamel’s Principle [545]
18.3 Further Discussion of Integrals [548]
18.4 The Integral as a Function of the Upper Limit [548]
18.41 The Integral of a Derivative [550]
18.5 Integrals Depending on a Parameter [551]
18.6 Riemann Double Integrals [554]
18.61 Double Integrals and Iterated Integrals [557]
18.7 Triple Integrals [559]
18.8 Improper Integrals [559]
18.9 Stieltjes Integrals [560]
19 INFINITE SERIES
19. Definitions and Notation [566]
19.1 Taylor’s Series [569]
19.11 A Series for the Inverse Tangent [572]
19.2 Series of Nonnegative Terms [573]
19.21 The Integral Test [577]
19.22 Ratio Tests [579]
19.3 Absolute and Conditional Convergence [581]
19.31 Rearrangement of Terms [585]
19.32 Alternating Series [587]
19.4 Tests for Absolute Convergence [590]
19.5 The Binomial Series [597]
19.6 Multiplication of Series [600]
19.7 Dirichlet’s Test [604]
20 UNIFORM CONVERGENCE
20. Functions Defined by Convergent Sequences [610]
20.1 The Concept of Uniform Convergence [613]
20.2 A Comparison Test for Uniform Convergence [618]
20.3 Continuity of the Limit Function [620]
20.4 Integration of Sequences and Series [621]
20.5 Differentiation of Sequences and Series [624]
21 POWER SERIES
21. General Remarks [627]
21.1 The Interval of Convergence [627]
21.2 Differentiation of Power Series [632]
21.3 Division of Power Series [639]
21.4 Abel’s Theorem [643]
21.5 Inferior and Superior Limits [647]
21.6 Real Analytic Functions [650]
22 IMPROPER INTEGRALS
22. Preliminary Remarks [654]
22.1 Positive Integrands. Integrals of the First Kind [656]
22.11 Integrals of the Second Kind [661]
22.12 Integrals of Mixed Type [664]
22.2 The Gamma Function [666]
22.3 Absolute Convergence [670]
22.4 Improper Multiple Integrals. Finite Regions [673]
22.41 Improper Multiple Integrals. Infinite Regions [678]
22.5 Functions Defined by Improper Integrals [682]
22.51 Laplace Transforms [690]
22.6 Repeated Improper Integrals [693]
22.7 The Beta Function [695]
22.8 Stirling’s Formula [699]
ANSWERS TO SELECTED EXERCISES [709]
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Incluye índice.

CONTENTS --
1 FUNDAMENTALS OF ELEMENTARY CALCULUS --
1. Introduction [1] --
1.1 Functions [2] --
1.11 Derivatives [12] --
1.12 Maxima and Minima [20] --
1.2 The Law of the Mean (The Mean-Value Theorem for Derivatives) [26] --
1.3 Differentials [32] --
1.4 The Inverse of Differentiation [35] --
1.5 Definite Integrals [38] --
1.51 The Mean-Value Theorem for Integrals [45] --
1.52 Variable Limits of Integration [46] --
1.53 The Integral of a Derivative [49] --
1.6 Limits [53] --
1.61 Limits of Functions of a Continuous Variable [54] --
1.62 Limits of Sequences [58] --
1.63 The Limit Defining a Definite Integral [67] --
1.64 The Theorem on Limits of Sums, Products, and Quotients [67] --
2 THE REAL NUMBER SYSTEM --
2. Numbers [72] --
2.1 The Field of Real Numbers [72] --
2.2 Inequalities. Absolute Value [74] --
2.3 The Principle of Mathematical Induction [75] --
2.4 The Axiom of Continuity [77] --
2.5 Rational and Irrational Numbers [78] --
2.6 The Axis of Reals [79] --
2.7 Least Upper Bounds [80] --
2.8 Nested Intervals [82] --
3 CONTINUOUS FUNCTIONS --
3. Continuity [85] --
3.1 Bounded Functions [86] --
3.2 The Attainment of Extreme Values [88] --
3.3 The Intermediate-Value Theorem [90] --
4 EXTENSIONS OF THE LAW OF THE MEAN --
4. Introduction [95] --
4.1 Cauchy’s Generalized Law of the Mean [95] --
4.2 Taylor’s Formula with Integral Remainder [97] --
4.3 Other Forms of the Remainder [99] --
4.4 An Extension of the Mean-Value Theorem for Integrals [105] --
4.5 L’Hospital’s Rule [106] --
5 FUNCTIONS OF SEVERAL VARIABLES --
5. Functions and Their Regions of Definition [116] --
5.1 Point Sets [117] --
5.2 Limits [122] --
5.3 Continuity [125] --
5.4 Modes of Representing a Function [127] --
6 THE ELEMENTS OF PARTIAL DIFFERENTIATION --
6. Partial Derivatives [130] --
6.1 Implicit Functions [132] --
6.2 Geometrical Significance of Partial Derivatives [135] --
6.3 Maxima and Minima [138] --
6.4 Differentials [144] --
6.5 Composite Functions and the Chain Rule [154] --
6.51 An Application in Fluid Mechanics [162] --
6.52 Second Derivatives by the Chain Rule [164] --
6.53 Homogeneous Functions. Euler’s Theorem [168] --
6.6 Derivatives of Implicit Functions [172] --
6.7 Extremal Problems with Constraints [177] --
6.8 Lagrange’s Method [182] --
6.9 Quadratic Forms [189] --
7 GENERAL THEOREMS OF PARTIAL DIFFERENTIATION --
7. Preliminary Remarks [196] --
7.1 Sufficient Conditions for Differentiability [197] --
7.2 Changing the Order of Differentiation [199] --
7.3 Differentials of Composite Functions [202] --
7.4 The Law of the Mean [204] --
7.5 Taylor’s Formula and Series [207] --
7.6 Sufficient Conditions for a Relative Extreme [211] --
8 IMPLICIT-FUNCTION THEOREMS --
8. The Nature of the Problem of Implicit Functions [222] --
8.1 The Fundamental Theorem [224] --
8.2 Generalization of the Fundamental Theorem [227] --
8.3 Simultaneous Equations [230] --
9 THE INVERSE FUNCTION THEOREM WITH APPLICATIONS --
9.Introduction [237] --
9.1 The Inverse Function Theorem in Two Dimensions [241] --
9.2 Mappings [247] --
9.3 Successive Mappings [252] --
9.4 Transformations of Co-ordinates [255] --
9.5 Curvilinear Co-ordinates [258] --
9.6 Identical Vanishing of the Jacobian. Functional Dependence [263] --
10 VECTORS AND VECTOR FIELDS --
10. Purpose of the Chapter [268] --
10.1 Vectors in Euclidean Space [268] --
10.11 Orthogonal Unit Vectors in R3 [273] --
10.12 The Vector Space Rn [274] --
10.2 Cross Products in R3 [280] --
10.3 Rigid Motions of the Axes [283] --
10.4 Invariants [286] --
10.5 Scalar Point Functions [291] --
10.51 Vector Point Functions [293] --
10.6 The Gradient of a Scalar Field [295] --
10.7 The Divergence of a Vector Field [300] --
10.8 The Curl of a Vector Field [305] --
11 I LINEAR TRANSFORMATIONS --
11. Introduction [309] --
11.1 Linear Transformations [312] --
11.2 The Vector Space L(Rn, Rm) [313] --
11.3 Matrices and Linear Transformations [313] --
11.4 Some Special Cases [316] --
11.5 Norms [318] --
11.6 Metrics [319] --
11.7 Open Sets and Continuity [320] --
11.8 A Norm on L(Rn,Rm) [324] --
11.9 L(Rn) [327] --
11.10 The Set of Invertible Operators [330] --
12 DIFFERENTIAL CALCULUS OF FUNCTIONS FROM Rn TO Rm --
12. Introduction [335] --
12.1 The Differential and the Derivative [336] --
12.2 The Component Functions and Differentiability [340] --
12.21 Directional Derivatives and the Method of Steepest Descent [343] --
12.3 Newton’s Method [347] --
12.4 A Form of the Law of the Mean for Vector Functions [350] --
12.41 The Hessian and Extreme Values [352] --
12.5 Continuously Differentiable Functions [354] --
12.6 The Fundamental Inversion Theorem [355] --
12.7 The Implicit Function Theorem [361] --
12.8 Differentiation of Scalar Products of Vector Valued Functions of a Vector Variable [366] --
13 DOUBLE AND TRIPLE INTEGRALS --
13. Preliminary Remarks [376] --
13.1 Motivations [376] --
13.2 Definition of a Double Integral [379] --
13.21 Some Properties of the Double Integral [381] --
13.22 Inequalities. The Mean-Value Theorem [382] --
13.23 A Fundamental Theorem [383] --
13.3 Iterated Integrals. Centroids [384] --
13.4 Use of Polar Co-ordinates [390] --
13.5 Applications of Double Integrals [395] --
13.51 Potentials and Force Fields [401] --
13.6 Triple Integrals [404] --
13.7 Applications of Triple Integrals [409] --
13.8 Cylindrical Co-ordinates [412] --
13.9 Spherical Co-ordinates [413] --
14 CURVES AND SURFACES --
14. Introduction [417] --
14.1 Representations of Curves [417] --
14.2 Arc Length [418] --
14.3 The Tangent Vector [421] --
14.31 Principal normal. Curvature [423] --
14.32 Binormal. Torsion [425] --
14.4 Surfaces [428] --
14.5 Curves on a Surface [433] --
14.6 Surface Area [437] --
15 LINE AND SURFACE INTEGRALS --
15. Introduction [445] --
15.1 Point Functions on Curves and Surfaces [445] --
15.12 Line Integrals [446] --
15.13 Vector Functions and Line Integrals. Work [451] --
15.2 Partial Derivatives at the Boundary of a Region [455] --
15.3 Green’s Theorem in the Plane [457] --
15.31 Comments on the Proof of Green’s Theorem [463] --
15.32 Transformations of Double Integrals [465] --
15.4 Exact Differentials [469] --
15.41 Line Integrals Independent of the Path [474] --
15.5 Further Discussion of Surface Area [478] --
15.51 Surface Integrals [480] --
15.6 The Divergence Theorem [484] --
15.61 Green’s Identities [492] --
15.62 Transformation of Triple Integrals [494] --
15.7 Stokes’s Theorem [499] --
15.8 Exact Differentials in Three Variables [505] --
16 POINT-SET THEORY --
16. Preliminary Remarks [512] --
16.1 Finite and Infinite Sets [512] --
16.2 Point Sets on a Line [514] --
16.3 The Bolzano-Weierstrass Theorem [517] --
16.31 Convergent Sequences on a Line [518] --
16.4 Point Sets in Higher Dimensions [520] --
16.41 Convergent Sequences in Higher Dimensions [521] --
16.5 Cauchy’s Convergence Condition [522] --
16.6 The Heine-Borel Theorem [523] --
17 FUNDAMENTAL THEOREMS ON CONTINUOUS FUNCTIONS --
17. Purpose of the Chapter [527] --
17.1 Continuity and Sequential Limits [527] --
17.2 The Boundedness Theorem [529] --
17.3 The Extreme-Value Theorem [529] --
17.4 Uniform Continuity [529] --
17.5 Continuity of Sums, Products, and Quotients [532] --
17.6 Persistence of Sign [532] --
17.7 The In termediate-Value Theorem [533] --
18 THE THEORY OF INTEGRATION --
18. The Nature of the Chapter [535] --
18.1 The Definition of Integrability [535] --
18.11 The Integrability of Continuous Functions [539] --
18.12 Integrable Functions with Discontinuities [540] --
18.2 The Integral as a Limit of Sums [542] --
18.21 Duhamel’s Principle [545] --
18.3 Further Discussion of Integrals [548] --
18.4 The Integral as a Function of the Upper Limit [548] --
18.41 The Integral of a Derivative [550] --
18.5 Integrals Depending on a Parameter [551] --
18.6 Riemann Double Integrals [554] --
18.61 Double Integrals and Iterated Integrals [557] --
18.7 Triple Integrals [559] --
18.8 Improper Integrals [559] --
18.9 Stieltjes Integrals [560] --
19 INFINITE SERIES --
19. Definitions and Notation [566] --
19.1 Taylor’s Series [569] --
19.11 A Series for the Inverse Tangent [572] --
19.2 Series of Nonnegative Terms [573] --
19.21 The Integral Test [577] --
19.22 Ratio Tests [579] --
19.3 Absolute and Conditional Convergence [581] --
19.31 Rearrangement of Terms [585] --
19.32 Alternating Series [587] --
19.4 Tests for Absolute Convergence [590] --
19.5 The Binomial Series [597] --
19.6 Multiplication of Series [600] --
19.7 Dirichlet’s Test [604] --
20 UNIFORM CONVERGENCE --
20. Functions Defined by Convergent Sequences [610] --
20.1 The Concept of Uniform Convergence [613] --
20.2 A Comparison Test for Uniform Convergence [618] --
20.3 Continuity of the Limit Function [620] --
20.4 Integration of Sequences and Series [621] --
20.5 Differentiation of Sequences and Series [624] --
21 POWER SERIES --
21. General Remarks [627] --
21.1 The Interval of Convergence [627] --
21.2 Differentiation of Power Series [632] --
21.3 Division of Power Series [639] --
21.4 Abel’s Theorem [643] --
21.5 Inferior and Superior Limits [647] --
21.6 Real Analytic Functions [650] --
22 IMPROPER INTEGRALS --
22. Preliminary Remarks [654] --
22.1 Positive Integrands. Integrals of the First Kind [656] --
22.11 Integrals of the Second Kind [661] --
22.12 Integrals of Mixed Type [664] --
22.2 The Gamma Function [666] --
22.3 Absolute Convergence [670] --
22.4 Improper Multiple Integrals. Finite Regions [673] --
22.41 Improper Multiple Integrals. Infinite Regions [678] --
22.5 Functions Defined by Improper Integrals [682] --
22.51 Laplace Transforms [690] --
22.6 Repeated Improper Integrals [693] --
22.7 The Beta Function [695] --
22.8 Stirling’s Formula [699] --
ANSWERS TO SELECTED EXERCISES [709] --

MR, 83m:26001

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