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 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