Advanced calculus / Angus E. Taylor, W. Robert Mann.
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 ContentsCONTENTS 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]
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 T238a (Browse shelf) | Checked out | 2024-04-01 | A-5413 |
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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