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Introduction to Real Analysis

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Introduction to Real Analysis

This book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.

AnalysisMathematics
Author
William F. Trench
Publisher
Trinity University
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 40 sections
  1. 01: The Real Numbers
  2. 1.00: The Real Number System
  3. 1.02: Mathematical Induction
  4. 1.03: The Real Line
  5. 02: Differential Calculus of Functions of One Variable
  6. 2.00: Functions and Limits
  7. 2.02: Continuity
  8. 2.03: Differentiable Functions of One Variable
  9. 2.04: L’Hospital’s Rule
  10. 2.05: Taylor’s Theorem
  11. 03: Integral Calculus of Functions of One Variable
  12. 3.00: Definition of the Integral
  13. 3.02: Existence of the Integral
  14. 3.03: Properties of the Integral
  15. 3.04: Improper Integrals
  16. 3.05: A More Advanced Look at the Existence of the Proper Riemann Integral
  17. 04: Infinite Sequences and Series
  18. 4.00: Sequences of Real Numbers
  19. 4.02: Earlier Topics Revisited With Sequences
  20. 4.03: Infinite Series of Constants
  21. 4.04: Sequences and Series of Functions
  22. 4.05: Power Series
  23. 05: Real-Valued Functions of Several Variables
  24. 5.00: Structure of Rn
  25. 5.02: Continuous Real-Valued Function of n Variables
  26. 5.03: Partial Derivatives and the Differential
  27. 5.04: The Chain Rule and Taylor’s Theorem
  28. 06: Vector-Valued Functions of Several Variables
  29. 6.00: Linear Transformations and Matrices
  30. 6.02: Continuity and Differentiability of Transformations
  31. 6.03: The Inverse Function Theorem
  32. 6.04: The Implicit Function Theorem
  33. 07: Integrals of Functions of Several Variables
  34. 7.00: Definition and Existence of the Multiple Integral
  35. 7.02: Iterated Integrals and Multiple Integrals
  36. 7.03: Change of Variables in Multiple Integrals
  37. 08: Metric Spaces
  38. 8.00: Introduction to Metric Spaces
  39. 8.02: Compact Sets in a Metric Space
  40. 8.03: Continuous Functions on Metric Spaces