Differentiability in Banach Spaces, Differential Forms and Applications

Description

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 final chapter contains an introduction to the Harmonic Functions and a geometric approach to Maxwell's equations of electromagnetism. 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 Self-adjoint Compact Operators on Hilbert Spaces 7.3 Fredholm Alternative 7.4 Hilbert-Schmidt 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 Euler-Lagrange 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 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 Λ(V) 2.1 Orientation 2.2 Inner Product in Λ(V) 2.3 Pseudo-Inner Product, the Lorentz Form 3 Differential Forms 3.1 Exterior Derivative 4 De Rham Cohomology 4.1 Short Exact Sequence 5 De Rham Cohomology of Spheres and Surfaces 6 Stokes Theorem 7 Orientation, Hodge Star-Operator and Exterior Co-derivative 8 Differential Forms on Manifolds, Stokes Theorem 8.1 Orientation 8.2 Integration on Manifolds 8.3 Exterior Derivative 8.4 Stokes Theorem on Manifolds 7 Applications to the Stokes Theorem 1 Volumes of the (n+1)-Disk and of the n-Sphere 2 Harmonic Functions 2.1 Laplacian Operator 2.2 Properties of Harmonic Functions 3 Poisson Kernel for the n-Disk DnR 4 Harmonic Differential Forms 4.1 Hodge Theorem on Manifolds 5 Geometric Formulation of the Electromagnetic Theory 5.1 Electromagnetic Potentials 5.2 Geometric Formulation 5.3 Variational Formulation 6 Helmholtz's Decomposition Theorem Appendix A Basics of Analysis 1 Sets 2 Finite-dimensional Linear Algebra: V=mathbbRn 2.1 Matrix Spaces 2.2 Linear Transformations 2.3 Primary Decomposition Theorem 2.4 Inner Product and Sesquilinear Forms 2.5 The Sylvester Theorem 2.6 Dual Vector Spaces 3 Metric and Banach Spaces 4 Calculus Theorems 4.1 One Real Variable Functions 4.2 Functions of Several Real Variables 5 Proper Maps 6 Equicontinuity and the Ascoli-Arzelà Theorem 7 Functional Analysis Theorems 7.1 Riesz and Hahn-Banach Theorems 7.2 Topological Complementary Subspace 8 The Contraction Lemma Appendix B Differentiable Manifolds, Lie Groups 1 Differentiable Manifolds 2 Bundles: Tangent and Cotangent 3 Lie Groups Appendix C Tensor Algebra 1 Tensor Product 2 Tensor Algebra Appendix References Index