Advanced calculus

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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