Advanced Calculus

Preface
Chapter 0. Preliminaries
0.1 Sets and Functions
0.2 Mathematical Induction
0.3 Countable and Uncountable Sets
0.4 Completeness of Real Numbers
0.5 Consequences of the Completeness
0.6 Problems

Part I. Functions of One Variable
Chapter 1. Sequences and Series
1.1. Limits and Sequences
1.2. Limit Theorems
1.3. The Bolzano-Weierstrass Theorem
1.4. Cauchy Sequences
1.5. Monotone Sequences
1.6. Limits Superior and Inferior
1.7. Series of Real Numbers
1.8. Convergence Tests for Series
1.9. Problems

Chapter 2. Limits and Continuity
2.1. Limits of Functions
2.2. Continuous Functions
2.3. Uniformly Continuous Functions
2.4. Monotone Functions
2.5. Problems

Chapter 3. Differentiation
3.1. Derivatives
3.2. Differentiation Rules
3.3. Exponential and Logarithmic Functions
3.4. Mean Value Theorem
3.5. L'Hospital’s Rule
3.6. Taylor’s Theorem
3.7. Problems

Chapter 4. Integration
4.1. Riemann Integrals
4.2. Properties of Integrals
4.3. Further Properties of Integrals
4.4. Fundamental Theorems of Calculus
4.5. Improper Integrals
4.6. Problems

Chapter 5. Sequences and Series of Functions
5.1. Double Series
5.2. Pointwise and Uniform Convergence
5.3. Consequences of Uniform Convergence
5.4. Power Series
5.5. Taylor Series
5.6. Problems

Part II. Functions of Several Variables
Chapter 6. Euclidean Spaces
6.1. The Euclidean Space
6.2. The Bolzano-Weierstrass Theorem in Real Vector Space
6.3. Open and Closed Sets
6.4. Compact Sets
6.5. Connected Sets
6.6. Problems

Chapter 7. Continuity of Multivariable Functions
7.1. Limit and Continuity in Real Vector Space
7.2. Properties of Continuous Functions
7.3. Contractions
7.4. Linear Functions
7.5. The Weierstrass Approximation Theorem
7.6. Problems

Chapter 8. Differentiation of Multivariable Functions
8.1. Partial Derivatives
8.2. Differentiability
8.3. Chain Rule
8.4. Mean Value Theorem in Real Vector Space
8.5. Inverse Function Theorem
8.6. Implicit Function Theorem
8.7. Optimization
8.8. Problems

Chapter 9. Integration of Multivariable Functions
9.1. Integrals on Hyperrectangles
9.2. Integrals on General Sets
9.3. Sets of Volume Zero and Integrable Functions
9.4. Iterated Integrals
9.5. Change of Variables: Preliminary Lemmas
9.6. Change of Variables
9.7. Evaluation of Some Integrals
9.8. Problems