Preface to the Second Edition
Preface to the First Edition
A Brief Introduction
Chapter 1 Euclidean Spaces
1 Smooth Functions on a Euclidean Space
1.1 C∞ Versus Analytic Functions
1.2 Taylor's Theorem with Remainder
Problems
2 Tangent Vectors in Rn as Derivations
2.1 The Directional Derivative
2.2 Germs of Functions
2.3 Derivations at a Point
2.4 Vector Fields
2.5 Vector Fields as Derivations
Problems
3 The Exterior Algebra of Multicovectors
3.1 Dual Space
3.2 Permutations
3.3 Multilinear Functions
3.4 The Permutation Action on Multilinear Functions
3.5 The Symmetrizing and Alternating Operators
3.6 The Tensor Product
3.7 The Wedge Product
3.8 Anticommutativity of the Wedge Product
3.9 Associativity of the Wedge Product
3.10 A Basis for k—Covectors
……
Chapter 2 Manifolds
Chapter 3 The Tangent Space
Chapter 4 Lie Groups and Lie Algebras
Chapter 5 Differential Forms
Chapter 6 Integration
Chapter 7 De Rham Theory
