دانلود کتاب Advanced Calculus

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PREFACE
CONTENTS
1 / Fundamentals of Elementary Calculus
1. Introduction
1.1 Functions
Exercises
1.11 Derivatives
Exercises
1.12 Maxima ana Minima
Exercises
1.2 The Law of the Mean (The Mean-Value Theorem for Derivatives)
Exercises
1.3 Differentials
Exercises
1.4 The Inverse of Differentiation
Exercises
1.5 Definite Integrals
Exercises
1.51 The Mean Value Theorem for Integrals
1.52 Variable Limits of Integration
1.53 The Integral of a Derivative
Exercises
1.6 Limits
1.61 Limits of Functions of a Continuous Variable
Exercises
1.62 Limits of Sequences
Exercises
1.63 The Limit Defining a Definite Integral
1.64 The Theorem on Limits of Sums, Products, and Quotients
Exercises
Miscellaneous Exercises
2 / The Real Number System
2. Numbers
2.1 The Field of Real Numbers
2.2 Inequalities. Absolute Value
Exercises
2.3 The Principle of Mathematical Induction
Exercises
2.4 The Axiom of Continuity
2.5 Rational and Irrational Numbers
Exercises
2.6 The Axis of Reals
2.7 Least Upper Bounds
Exercises
2.8 Nested Intervals
Miscellaneous Exercises
3 / Continuous Functions
3. Continuity
Exercises
3.1 Bounded Functions
Exercises
3.2 The Attainment of Extreme Values
Exercises
3.3 The Intermediate-Value Theorem
Exercises
Miscellaneous Exercises
4 / Extensions of the Law of the Mean
4. Introduction
4.1 Cauchy\'s Generalized Law of the Mean
Exercises
4.2 Taylor\'s Formula with Integral Remainder
4.3 Other Forms of the Remainder
Exercises
4.4 An Extension of the Mean-Value Theorem for Integrals
4.5 L\'Hospital\'s Rule
Exercises
Miscellaneous Exercises
5 / Functions of Several Variables
5. Functions and Their Regions of Definition
5.1 Point Sets
Exercises
5.2 Limits
Exercises
5.3 Continuity
Exercises
5.4 Modes of Representing a Function
6 / The Elements of Partial Differentiation
6. Partial Derivatives
6.1 Implicit Functions
Exercises
6.2 Geometrical Significance of Partial Derivatives
Exercises
6.3 Maxima and Minima
Exercises
6.4 Differentials
Exercises
6.5 Composite Functions and the Chain Rule
Exercises
6.51 An Application in Fluid Mechanics
Exercises
6.52 Second Derivatives by the Chain Rule
Exercises
6.53 Homogeneous Functions. Euler\'s Theorem
Exercises
6.6 Derivatives of Implicit Functions
Exercises
6.7 Extremal Problems with Constraints
6.8 Lagrange\'s Method
Exercises
6.9 Quadratic Forms
Exercises
Miscellaneous Exercises
7 / General Theorems of Partial Differentiation
7. Preliminary Remarks
7.1 Sufficient Conditions for Differentiability
Exercises
7.2 Changing the Order of Differentiation
Exercises
7.3 Differentials of Composite Functions
7.4 The Law of the Mean
Exercises
7.5 Taylor\'s Formula and Series
Exercises
7.6 Sufficient Conditions for a Relative Extreme
Exercises
Miscellaneous Exercises
8 / Implicit-Function Theorems
8. The Nature of the Problem of Implicit Functions
8.1 The Fundamental Theorem
8.2 Generalization of the Fundamental Theorem
Exercises
8.3 Simultaneous Equations
Exercises
9 / The Inverse Function Theorem with Applications
9. Introduction
9.1 The Inverse Function Theorem in Two Dimensions
Exercise
9.2 Mappings
Exercises
9.3 Successive Mappings
Exercises
9.4 Transformations of Co-ordinates
9.5 Curvilinear Co-ordinates
Exercises
9.6 Identical Vanishing of the Jacobian. Functional Dependence
Exercises
Miscellaneous Exercises
10 / Vectors and Vector Fields
10. Purpose of the Chapter
10.1 Vectors in Euclidean Space
10.11 Orthogonal Unit Vectors in ℝ³
Exercises
10.12 The Vector Space ℝⁿ
Exercises
10.2 Cross Products in ℝ³
Exercises
10.3 Rigid Motions of the Axes
Exercises
10.4 Invariants
Exercises
10.5 Scalar Point Functions
10.51 Vector Point Functions
10.6 The Gradient of a Scalar Field
Exercises
10.7 The Divergence of a Vector Field
Exercises
10.8 The Curl of a Vector Field
Exercises
Miscellaneous Exercises
11 / Linear Transformations
11. Introduction
11.1 Linear Transformations
11.2 The Vector Space ℒ(ℝⁿ, ℝᵐ)
11.3 Matrices and Linear Transformations
11.4 Some Special Cases
11.5 Norms
11.6 Metrics
11.7 Open Sets and Continuity
11.8 A Norm on ℒ(ℝⁿ, ℝᵐ)
11.9 ℒ(ℝⁿ)
11.10 The Set of Invertible Operators
Exercises
12 / Differential Calculus of Functions from ℝⁿ to ℝᵐ
12. Introduction
12.1 The Differential and the Derivative
12.2 The Component Functions and Differentiability
12.21 Directional Derivatives and the Method of Steepest Descent
12.3 Newton\'s Method
12.4 A Form of the Law of the Mean for Vector Functions
12.41 The Hessian and Extreme Values
12.5 Continuously Differentiable Functions
12.6 The Fundamental Inversion Theorem
12.7 The Implicit Function Theorem
12.8 Differentiation of Scalar Products of Vector Valued Functions of a Vector Variable
Exercises
13 / Double and Triple Integrals
13. Preliminary Remarks
13.1 Motivations
13.2 Definition of a Double Integral
13.21 Some Properties of the Double Integral
13.22 Inequalities. The Mean-Value Theorem
13.23 A Fundamental Theorem
13.3 Iterated Integrals. Centroids
Exercises
13.4 Use of Polar Co-ordinates
Exercises
13.5 Applications of Double Integrals
Exercises
13.51 Potentials and Force Fields
Exercises
13.6 Triple Integrals
13.7 Applications of Triple Integrals
Exercises
13.8 Cylindrical Co-ordinates
Exercises
13.9 Spherical Co-ordinates
Exercises
14 / Curves and Surfaces
14. Introduction
14.1 Representations of Curves
14.2 Arc Length
Exercises
14.3 The Tangent Vector
Exercises
14.31 Principal normal. Curvature
14.32 Binormal. Torsion
Exercises
14.4 Surfaces
Exercises
14.5 Curves on a Surface
Exercises
14.6 Surface Area
Exercises
15 / Line and Surface Integrals
15. Introduction
15.1 Point Functions on Curves and Surfaces
15.12 Line Integrals
Exercises
15.13 Vector Functions and Line Integrals. Work
Exercises
15.2 Partial Derivatives at the Boundary of a Region
15.3 Green\'s Theorem in the Plane
Exercises
15.31 Comments on the Proof of Green\'s Theorem
15.32 Transformations of Double Integrals
Exercises
15.4 Exact Differentials
15.41 Line Integrals Independent of the Path
Exercises
15.5 Further Discussion of Surface Area
15.51 Surface Integrals
Exercises
15.6 The Divergence Theorem
Exercises
15.61 Green\'s Identities
Exercises
15.62 Transformation of Triple Integrals
Exercises
15.7 Stokes\'s Theorem
Exercises
15.8 Exact Differentials in Three V ariables
Exercises
Miscellaneous Exercises
16 / Point-Set Theory
16. Preliminary Remarks
16.1 Finite and Infinite Sets
16.2 Point Sets on a Line
Exercises
16.3 The Bolzano-Weierstrass Theorem
Exercises
16.31 Convergent Sequences on a Line
Exercises
16.4 Point Sets in Higher Dimensions
16.41 Convergent Sequences in Higher Dimensions
Exercises
16.5 Cauchy\'s Convergence Condition
16.6 The Heine-Borel Theorem
Exercises
17 / Fundamental Theorems on Continious Functions
17. Purpose of the Chapter
17.1 Continuity and Sequential Limits
17.2 The Boundedness Theorem
17.3 The Extreme-Value Theorem
17.4 Uniform Continuity
17.5 Continuity of Sums, Products, and Quotients
Exercises
17.6 Persistence of Sign
17.7 The Intermediate-Value Theorem
18 / The Theory of Integration
18. The Nature of the Chapter
18.1 The Definition of Integrability
Exercises
18.11 The Integrability of Continuous Functions
Exercise
18.12 Integrable Functions with Discontinuities
18.2 The Integral as a Limit of Sums
Exercises
18.21 Duhamel\'s Principle
Exercises
18.3 Further Discussion of Integrals
18.4 The Integral as a Function of the U pper Limit
Exercises
18.41 The Integral of a Derivative
18.5 Integrals Depending on a Parameter
Exercises
18.6 Riemann Double Integrals
Exercises
18.61 Double Integrals and Iterated Integrals
18.7 Triple Integrals
18.8 Improper Integrals
18.9 Stieltjes Integrals
Exercises
19 / Infinite Series
19. Definitions and Notation
Exercises
19.1 Taylor\'s Series
Exercises
19.11 A Series for the Inverse Tangent
Exercises
19.2 Series of Nonnegative Terms
Exercises
19.21 The Integral Test
Exercises
19.22 Ratio Tests
Exercises
19.3 Absolute and Conditional Convergence
Exercises
19.31 Rearrangement of Terms
Exercises
19.32 Alternating Series
Exercises
19.4 Tests for Absolute Convergence
Exercises
19.5 The Binomial Series
Exercises
19.6 Multiplication of Series
Exercises
19.7 Dirichlet\'s Test
Exercises
Miscellaneous Exercises
20 / Uniform Convergence
20. Functions Defined by Convergent Sequences
20.1 The Concept of Uniform Convergence
Exercises
20.2 A Comparison Test for Uniform Convergence
Exercises
20.3 Continuity of the Limit Function
Exercises
20.4 Integration of Sequences and Series
Exercises
20.5 Differentiation of Sequences and Series
Exercises
21 / Power Series
21. General Remarks
21.1 The Interval of Convergence
Exercises
21.2 Differentiation of Power Series
Exercises
21.3 Division of Power Series
Exercises
21.4 Abel\'s Theorem
Exercises
21.5 Inferior and Superior Limits
Exercises
21.6 Real Analytic Functions
Exercises
Miscellaneous Exercises
22 / Improper Integrals
22. Preliminary Remarks
22.1 Positive Integrands. Integrals of the First Kind
Exercises
22.11 Integrals of the Second Kind
Exercises
22.12 Integrals of Mixed Type
Exercises
22.2 The Gamma Function
Exercises
22.3 Absolute Convergence
Exercises
22.4 Improper Multiple Integrals. Finite Regions
Exercises
22.41 Improper Multiple Integrals. Infinite Regions
Exercises
22.5 Functions Defined by Improper Integrals
Exercises
22.51 Laplace Transforms
Exercises
22.6 Repeated Improper Integrals
Exercises
22.7 The Beta Function
Exercises
22.8 Stirling\'s Formula
Miscellaneous Exercises
Answers to Selected Exercises
Answers 1.1-1.2
Answers 1.3-2.7
Answers 2.Mis.-3.Mis
Answers 4.3-5.3
Answers 6.1-6.8
Answers 6.9-7.6
Answers 7.Mis-9.5
Answers 9.6-10.8
Answers 10.Mis-13.5
Answers 13.5-14.32
Answers 14.32-15.13.
Answers 15.3-15.8
Answers 15.Mis-18.6
Answers 18.9-19.4
Answers 19.4-21.2
Answers 21.2-22.2
Answers 22.3-END
Index
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