Differentiability in Banach Spaces, Differ...

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Beschreibung

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.

Über den Autor

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.

Zusammenfassung

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

Inhaltsverzeichnis

Introduction.- Chapter 1. Differentiation in R^n.- Chapter 2. Linear Operators in Banach Spaces.- Chapter 3. Differentiation in Banach Spaces.- Chapter 4. Vector Fields.- Chapter 5. Vectors Integration, Potential Theory.- Chapter 6. Differential Forms, Stoke's Theorem.- Chapter 7. Applications to the Stoke's Theorem.- Appendix A. Basics of Analysis.- Appendix B. Differentiable Manifolds, Lie Groups.- Appendix C. Tensor Algebra.- Bibliography.- Index.

Erscheinungsjahr:	2021
Fachbereich:	Analysis
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Inhalt:	xiv 362 S. 43 s/w Illustr. 26 farbige Illustr. 362 p. 69 illus. 26 illus. in color.
ISBN-13:	9783030778330
ISBN-10:	3030778339
Sprache:	Englisch
Ausstattung / Beilage:	HC runder Rücken kaschiert
Einband:	Gebunden
Autor:	Doria, Celso Melchiades
Auflage:	1st ed. 2021
Hersteller:	Springer International Publishing
Maße:	241 x 160 x 26 mm
Von/Mit:	Celso Melchiades Doria
Erscheinungsdatum:	20.07.2021
Gewicht:	0,729 kg

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