Differentiability in Banach Spaces, Differential Forms and Applications

1. Auflage 2021
ISBN: 978-3-030-77836-1
Verlag: Springer International Publishing

Buch, Englisch, 362 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 569 g

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.

Zielgruppe

Graduate

Autoren/Hrsg.

Doria, Celso Melchiades

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure

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.

Über Autor(innen)

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.