Differentiability in Banach Spaces, Differential Forms and Applications

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