Differentiability in Banach spaces, differential forms and applications

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.

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…