Real Analysis and Applications

1. Front Matter

   Pages i-xiii

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2. One Variable Real Analysis

   1. Front Matter

      Pages 1-1

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   2. Real Numbers

      • Fabio Silva Botelho

      Pages 3-49

   3. Metric Spaces

      • Fabio Silva Botelho

      Pages 51-64

   4. Real Sequences and Series

      • Fabio Silva Botelho

      Pages 65-96

   5. Real Function Limits

      • Fabio Silva Botelho

      Pages 97-140

   6. Continuous Functions

      • Fabio Silva Botelho

      Pages 141-185

   7. Derivatives

      • Fabio Silva Botelho

      Pages 187-209

   8. The Riemann Integral

      • Fabio Silva Botelho

      Pages 211-270

3. Multi-Variable Advanced Calculus

   1. Front Matter

      Pages 271-271

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   2. Differential Analysis in \(\mathbb {R}^n\)

      • Fabio Silva Botelho

      Pages 273-416

   3. Integration in \(\mathbb {R}^n\)

      • Fabio Silva Botelho

      Pages 417-469

   4. Topics on Vector Calculus and Vector Analysis in \(\mathbb {R}^n\)

      • Fabio Silva Botelho

      Pages 471-558

4. Back Matter

   Pages 559-567

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This textbook introduces readers to real analysis in one and n dimensions. It is divided into two parts: Part I explores real analysis in one variable, starting with key concepts such as the construction of the real number system, metric spaces, and real sequences and series. In turn, Part II addresses the multi-variable aspects of real analysis. Further, the book presents detailed, rigorous proofs of the implicit theorem for the vectorial case by applying the Banach fixed-point theorem and the differential forms concept to surfaces in Rn. It also provides a brief introduction to Riemannian geometry. 
With its rigorous, elegant proofs, this self-contained work is easy to read, making it suitable for undergraduate and beginning graduate students seeking a deeper understanding of real analysis and applications, and for all those looking for a well-founded, detailed approach to real analysis.

• real analysis
• Implicit function theorem
• Banach's fixed point theorem
• metric spaces
• Arzela-Ascoli theorem
• Riemannian geometry

• Department of Mathematics, Federal University of Santa Catarina, Florianópolis, Brazil

  Fabio Silva Botelho

Fabio Botelho holds a PhD in Mathematics from Virginia Tech, USA, and a Master in Aeronautics and Mechanics Engineering from the Aeronautics Institute of Technology, Brazil. He is the author of the book "Functional Analysis and Applied Optimization in Banach Spaces," also published with Springer. His main research fields are calculus of variations, convex analysis and duality applied to problems in physics and engineering.

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