目錄
Preface to the Third Edition
Preface to the Second Edition
Preface to the Revised Printing
Preface to the First Edition
Overview. An Informal Overview of Cartan's Exterior Differential Forms, Illustrated with an Application to Cauchy's Stress Tensor
Introduction
O.a.Introduction
Vectors,1-Forms,and Tensors
0.b.Two Kinds of Vectors
0.c.Superscripts,Subscripts,Summation Convention
0.d.Riemannian Metrics
0.e.Tensors
Integrals and Exterior Forms
0.f.Line Integrals
0.g.Exterior 2-Forms
0.h.Exterior p-Forms and Algebra in Rn
0.i.The Exterior Differential d
0.j.The Push-Forward of a Vector and the Pull-Back of a Form
o.k.Surface Integrals and「Stokes'Theorem」
0.1.Electromagnetism,or,Is it a Vector or a Form?
C.m.Interior Products
o.n.Volume Forms and Cartan's Vector Valued Exterior Forms
0.o.Magnetic Field for Curent in a Straight Wire
Elasticity and Stresses
0.p.Cauchy Stress,Floating Bodies,Twisted Cylinders,and Strain Energy
0.q.Sketch of Cauchy's「First Theorem」
0.r.Sketch of Cauchy's「Second Theorem,」Moments as Generators of Rotations
0.s. ARemarkable Formula for Differentiating Line,Surface. and...,Integrals
Ⅰ Manifolds,Tensors,and Exterior Forms
1 Manifolds and Vector Fields
1.1.Submanifolds of Euclidean Space
1.1a.Submanifolds of RN
1.1b.The Geometry of Jacobian Matrices: The 「Differential」
1.1c.The Main Theorem on Submanifolds of RN
1.1d.A Nontrivial Example: The Configuration Space of a Rigid Body
1.2.Manifolds
1.2a.Some Notions from Point Set Topology
1.2b.The Idea of a Manifold
1.2c.A Rigorous Definition of a Manifold
1.2d.Complex Manifolds:The Riemann Sphere
1.3.Tangent Vectors and Mappings
1.3a.Tangent or 「Contravariant」Vectors
1.3b.Vectors as Differential Operators
1.3c.The Tangent Space to Mn at a Point
1.3d.Mappings and Submanifolds of Manifolds
1.3e.Change of Coordinates
1.4.Vector Fields and Flows
1.4a.Vector Fields and Flows on Rn
1.4b.Vector Fields on Manifolds
1.4c.Straightening Flows
2 Tensors and Exterior Forms
2.1.Covectors and Riemannian Metrics
2.1a.Linear Functionals and the Dual Space
2.1b.The Differential of a Function
2.1c.Scalar Products in Linear Algebra
2.1d.Riemannian Manifolds and the Gradient Vector
2.1e.Curves of Steepest Ascent
2.2.The Tangent Bundle
2.2a.The Tangent Bundle
2.2b.The Unit Tangent Bundle
2.3.The Cotangent Bundle and Phase Space
2.3a.The Cotangent Bundle
2.3b.The Pull-Back of a Covector
2.3c.The Phase Space in Mechanics
2.3d.The Poincare 1-Form
2.4.Tensors
2.4a.Covariant Tensors
2.4b.Contravariant Tensors
2.4c.Mixed Tensors
2.4d.Transformation Properties of Tensors
2.4e.Tensor Fields on Manifolds
……
Ⅱ Geometry and Topology
Ⅲ Lie Groups, Bundles, and Chern Forms
References
Index
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