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Introduction to Mathematical Analysis

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1 The Structure of the Real Numbers: Sequences
1.1 Completeness of the Real Numbers
1.2 Neighborhood and Limit Points
1.3 The Limit of a Sequence
1.4 Cauchy Sequences
1.5 The Algebra of Convergent Sequuences
1.6 Cardinlity

2 Euclidean Spaces
2.1 Euclidean n-Spaces
2.2 Open and Closed Sets
2.3 Completeness
2.4 Relative Topology and Connectedness
2.5 Compactness

3 Continuity
3.1 Limit and Continuty
3.2 The Topological Descroption of Continuity
3.3 The Algebra of Contunuous Functions
3.4 Uniform Continuity
3.5 The Uniform Norm: Uniform Convergence
3.6 Vector-Valued Function on IRn

4 Differentiation
4.1 The Derivative
4.2 Composition of Function: The Chain Rule
4.3 The Mean Value Theorem
4.4 L'hopital's Rule
4.5 Taylor's Theorem

5 Functions of Bounded Variations
5.1 Partitions
5.2 Monotone Functions on [a, b]
5.3 Functions of Bounded Variations
5.4 Total Variation as a Function
5.5 Continuous Fuctions of Bounded Variation

6 The Riemann Integral
6.1 Definition of the Riemann Integral
6.2 Existence of the Riemann Integral
6.3 The Fundamental Theorem of Caculus
6.4 Techniques of Integration
6.5 Uniform Convergence and the Integral Exercises

7 The Riemann-Stiltjes Integral
7.1 Definition of the Riemann-Stieltjes Integral
7.2 Techniques of Integration
7.3 Existence of the Riemann-Stieltjes Integral
7.4 Fundmental Theorems of the Riemann-Stieltjes Integration
Exercises

8 Differential Calculus in IRn
8.1 Differentiability
8.2 The Algebra of Differentiable Functions
8.3 Differentiability of Vector-Valued Functions
8.4 The Chain Rule
8.5 The Mean Value Theorem
8.6 Higher-Order Partial Derivatives
8.7 Taylor's Theorem
8.8 Extreme Values of Differentiable Functions
Execises

9 Vector-Valued Functions
9.1 The Joacobian
9.2 The Inverse Function Theorem
9.3 The Implicit Function Theorem
9.4 Constrained Optimization
Exercises

10 Multiple Integrals
10.1 The Double Integral
10.2 Evaluation of Double Integrals
10.3 Transformations: Change of Variables
10.4 Multiple Integrals in IR3
Exercises

11 Infinite Series
11.1 Preliminaries
11.2 Convergence Test (Positive Series)
11.3 Absolute Convergence
11.4 Conditional Convergence
11.5 The Cauchy Product
11.6 Cesaro Summability
Execrises

12 Series of Functions
12.1 Preliminaries
12.2 Uniform Convegence
12.3 Tests for Uniform Convergence
12.4 Power Series
12.5 The Taylor Series Representation of Function
12.6 Solutions of First-Order Differential Equations
Exercises

13 Improper Integral
13.1 Preliminaries
13.2 Improper Integrals of the First Kind
13,3 Improper Integrals of the Second Kind
13.4 Uniform Convergence of Improper Integrals
13.5 Functions Defined by Improper Integrals
13.6 The Laplace Tranform
Exercises

14 Fourier Series
14.1 Covergence of the Mean
14.2 Trigonometric Series
14.3 Convergence of Trigonmetric Series
14.4 The Cesaro Summability of Fourier Series
14.5 Addtional Topics
Exercises

Axioms for the Real NumBers IR
Set Theory
Functions
Polynomials
References and Additional Readings
Index

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