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Differentiability in Banach Spaces, Differential Forms and Applications

  • Textbook
  • © 2021

Overview

Authors:
  1. Celso Melchiades Doria

    1. Department of Mathematics, CFM, Universidade Federal de Santa Catarina, Florianópolis, Brazil

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  • The differential forms formalism is explained through the classical theorems of integrations and applied to obtain topological invariants
  • Includes applications to the study of harmonic functions and to the formulation of the Maxwell’s equations using differential forms
  • Avoiding complicated notation

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Differentiation in \(\mathbb {R}^{n}\)

    • Celso Melchiades Doria
    Pages 1-76

  3. Linear Operators in Banach Spaces

    • Celso Melchiades Doria
    Pages 77-125

  4. Differentiation in Banach Spaces

    • Celso Melchiades Doria
    Pages 127-178

  5. Vector Fields

    • Celso Melchiades Doria
    Pages 179-213

  6. Vector Integration, Potential Theory

    • Celso Melchiades Doria
    Pages 215-230

  7. Differential Forms, Stokes Theorem

    • Celso Melchiades Doria
    Pages 231-280

  8. Applications to the Stokes Theorem

    • Celso Melchiades Doria
    Pages 281-315

  9. Back Matter

    Pages 317-362
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Keywords

About this book

This book is divided into two parts, the first one to study the theory of differentiable functions between Banach spaces and the second to study the differential form formalism and to address the Stokes' Theorem and its applications. Related to the first part, there is an introduction to the content of Linear Bounded Operators in Banach Spaces with classic examples of compact and Fredholm operators, this aiming to define the derivative of Fréchet and to give examples in Variational Calculus and to extend the results to Fredholm maps. The Inverse Function Theorem is explained in full details to help the reader to understand the proof details and its motivations. The inverse function theorem and applications make up this first part. The text contains an elementary approach to Vector Fields and Flows, including the Frobenius Theorem. The Differential Forms are introduced and applied to obtain the Stokes Theorem and to define De Rham cohomology groups. As an application, the finalchapter contains an introduction to the Harmonic Functions and a geometric approach to Maxwell's equations of electromagnetism.




Reviews

“A specific feature of the book is the abundance of examples from mechanics, physics, calculus of variations, illustrating the abstract concepts introduced in the main text. … There are a lot of exercises spread through the book, some elementary, while others are more advanced. The book can be used as supplementary material for undergraduate or graduate level courses, as well as by the students in physics interested in a mathematical treatment of some important problems in their domain.” (Stefan Cobzaş, zbMATH 1479.46001, 2022)

Authors and Affiliations

  • Department of Mathematics, CFM, Universidade Federal de Santa Catarina, Florianópolis, Brazil

    Celso Melchiades Doria

About the author

The author is Professor of Mathematics at the Universidade Federal de Santa Catarina where he is a faculty member since 1993. He holds a PhD title in Mathematics from the University of Warwick, England, under the supervision of Professor James Eells. His research interest lies on Global Analysis, concentrating on the geometry of Gauge Fields and its applications to the Topology and to the Geometry of differentiable manifolds. His scientific background includes a postdoctoral at the Mathematical Institute, Oxford University, England, and another at Michigan State University, USA.

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