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- Preface
- Summary of Each Chapter Content
- Chapter 1: Real Numbers
- Chapter 2: Metric Spaces
- Chapter 3: Real Sequences and Series
- Chapter 4: Real Function Limits
- Chapter 5: Continuous Functions
- Chapter 6: Derivatives
- Chapter 7: The Riemann Integral
- Chapter 8: Differential Analysis in Rn
- Chapter 9: Integration in Rn
- Chapter 10: Topics on Vector Calculus and Vector Analysis in Rn
- Acknowledgments
- Summary of Each Chapter Content
- Contents
- Part I One Variable Real Analysis
- 1 Real Numbers
- 2 Metric Spaces
- 3 Real Sequences and Series
- 4 Real Function Limits
- 5 Continuous Functions
- 6 Derivatives
- 7 The Riemann Integral
- Part II Multi-Variable Advanced Calculus
- 8 Differential Analysis in Rn
- 8.1 Introduction
- 8.2 The Space Rn
- 8.3 Topology for Rn
- 8.4 Scalar Functions of Several Variables
- 8.5 Limits
- 8.6 Continuous Functions
- 8.7 Partial Derivatives
- 8.8 Differentiability in Rn
- 8.9 More Details About the Differentiability in R2
- 8.10 The Chain Rule in Rn
- 8.11 Higher Order Partial Derivatives
- 8.12 The Taylor Theorem for Rn
- 8.13 Local and Global Extremal Points for Functions in Rn
- 8.14 Critical Points
- 8.15 The Implicit Function Theorem
- 8.16 Implicit Function Theorem, Scalar Case
- 8.17 Vectorial Functions in Rn
- 8.18 Implicit Function Theorem for the Vectorial Case
- 8.19 Lagrange Multipliers
- 8.20 Lagrange Multipliers: The General Case
- 8.21 Inverse Function Theorem
- 8.22 Some Topics on Differential Geometry
- 8.23 Some Notes About the Scalar Curvature in a Surface in R3
- 9 Integration in Rn
- 10 Topics on Vector Calculus and Vector Analysis in Rn
- 10.1 Curves in R3
- 10.2 Line Integrals
- 10.3 On the Green Theorem in R2
- 10.4 Exact Differential Forms
- 10.5 Surface Integrals
- 10.6 Divergence Theorem
- 10.7 The Stokes Theorem in R3
- 10.8 Surfaces in Rn
- 10.9 The Tangent Space
- 10.10 Vector Fields
- 10.11 The Generalized Derivative
- 10.12 Differential Forms
- 10.13 Integration of Differential Forms
- 10.14 A Simple Example to Illustrate the Integration Process
- 10.15 Volume (Area) of a Surface
- 10.16 Change of Variables: The General Case
- 10.17 The Stokes Theorem
- 10.18 The First and Second Green Identities in a Surface
- 10.19 More Topics on Differential Geometry
- 10.20 Some Topics on Riemannian Geometry
- 8 Differential Analysis in Rn
- References
- Index
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Real Analysis and Application
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