Description 
1 online resource 
Contents 
5 Vector Fields as Differential Operators  6 Integrability, Frobenius Theorem  7 Lie Groups and Lie Algebras  8 Variations over a Flow, Lie Derivative  9 Gradient, Curl and Divergent Differential Operators  5 Vector Integration, Potential Theory  1 Vector Calculus  1.1 Line Integral  1.2 Surface Integral  2 Classical Theorems of Integration  2.1 Interpretation of the Curl and Div Operators  3 Elementary Aspects of the Theory of Potential  6 Differential Forms, Stokes Theorem  1 Exterior Algebra  2 Orientation on V and on the Inner Product on [delta](V)  2.1 Orientation 

Intro  Preface  Introduction  Contents  1 Differentiation in mathbbRn  1 Differentiability of Functions f:mathbbRnrightarrowmathbbR  1.1 Directional Derivatives  1.2 Differentiable Functions  1.3 Differentials  1.4 Multiple Derivatives  1.5 Higher Order Differentials  2 Taylor's Formula  3 Critical Points and Local Extremes  3.1 Morse Functions  4 The Implicit Function Theorem and Applications  5 Lagrange Multipliers  5.1 The Ultraviolet Catastrophe: The Dawn of Quantum Mechanics  6 Differentiable Maps I  6.1 Basics Concepts  6.2 Coordinate Systems 

6.3 The Local Form of an Immersion  6.4 The Local Form of Submersions  6.5 Generalization of the Implicit Function Theorem  7 Fundamental Theorem of Algebra  8 Jacobian Conjecture  8.1 Case n=1  8.2 Case nge2  8.3 Covering Spaces  8.4 Degree Reduction  2 Linear Operators in Banach Spaces  1 Bounded Linear Operators on Normed Spaces  2 Closed Operators and Closed Range Operators  3 Dual Spaces  4 The Spectrum of a Bounded Linear Operator  5 Compact Linear Operators  6 Fredholm Operators  6.1 The Spectral Theory of Compact Operators 

7 Linear Operators on Hilbert Spaces  7.1 Characterization of Compact Operators on Hilbert Spaces  7.2 Selfadjoint Compact Operators on Hilbert Spaces  7.3 Fredholm Alternative  7.4 HilbertSchmidt Integral Operators  8 Closed Unbounded Linear Operators on Hilbert Spaces  3 Differentiation in Banach Spaces  1 Maps on Banach Spaces  1.1 Extension by Continuity  2 Derivation and Integration of Functions f:[a, b]rightarrowE  2.1 Derivation of a Single Variable Function  2.2 Integration of a Single Variable Function  3 Differentiable Maps II 

4 Inverse Function Theorem (InFT)  4.1 Prelude for the Inverse Function Theorem  4.2 InFT for Functions of a Single Real Variable  4.3 Proof of the Inverse Function Theorem (InFT)  4.4 Applications of InFT  5 Classical Examples in Variational Calculus  5.1 EulerLagrange Equations  5.2 Examples  6 Fredholm Maps  6.1 Final Comments and Examples  7 An Application of the Inverse Function Theorem to Geometry  4 Vector Fields  1 Vector Fields in mathbbRn  2 Conservative Vector Fields  3 Existence and Uniqueness Theorem for ODE  4 Flow of a Vector Field 
Summary 
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 Frechet 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 final chapter contains an introduction to the Harmonic Functions and a geometric approach to Maxwell's equations of electromagnetism 
Bibliography 
Includes bibliographical references and index 
Notes 
Online resource; title from PDF title page (SpringerLink, viewed August 9, 2021) 
Subject 
Banach spaces.


Stokes' theorem.


Banach spaces


Stokes' theorem


Espais de Banach.

Genre/Form 
Llibres electrĂ˛nics.

Form 
Electronic book

ISBN 
9783030778347 

3030778347 
