Preface |
Motivation and Background / Part 1: |
The Case for Differential Geometry / 1: |
Classical Space-Time and Fibre Bundles / 1.1: |
Configuration Manifolds and Their Tangent and Cotangent Spaces / 1.2: |
The Infinite-dimensional Case / 1.3: |
Elasticity / 1.4: |
Material or Configurational Forces / 1.5: |
Vector and Affine Spaces / 2: |
Vector Spaces: Definition and Examples / 2.1: |
Linear Independence and Dimension / 2.2: |
Change of Basis and the Summation Convention / 2.3: |
The Dual Space / 2.4: |
Linear Operators and the Tensor Product / 2.5: |
Isomorphisms and Iterated Dual / 2.6: |
Inner-product Spaces / 2.7: |
Affine Spaces / 2.8: |
Banach Spaces / 2.9: |
Tensor Algebras and Multivectors / 3: |
The Algebra of Tensors on a Vector Space / 3.1: |
The Contravariant and Covariant Subalgebras / 3.2: |
Exterior Algebra / 3.3: |
Multivectors and Oriented Affine Simplexes / 3.4: |
The Faces of an Oriented Affine Simplex / 3.5: |
Multicovectors or r-Forms / 3.6: |
The Physical Meaning of r-Forms / 3.7: |
Some Useful Isomorphisms / 3.8: |
Differential Geometry / Part 2: |
Differentiable Manifolds / 4: |
Introduction / 4.1: |
Some Topological Notions / 4.2: |
Topological Manifolds / 4.3: |
Differentiability / 4.4: |
Tangent Vectors / 4.6: |
The Tangent Bundle / 4.7: |
The Lie Bracket / 4.8: |
The Differential of a Map / 4.9: |
Immersions, Embeddings, Submanifolds / 4.10: |
The Cotangent Bundle / 4.11: |
Tensor Bundles / 4.12: |
Pull-backs / 4.13: |
Exterior Differentiation of Differential Forms / 4.14: |
Some Properties of the Exterior Derivative / 4.15: |
Riemannian Manifolds / 4.16: |
Manifolds with Boundary / 4.17: |
Differential Spaces and Generalized Bodies / 4.18: |
Lie Derivatives, Lie Groups, Lie Algebras / 5: |
The Fundamental Theorem of the Theory of ODEs / 5.1: |
The Flow of a Vector Field / 5.3: |
One-parameter Groups of Transformations Generated by Flows / 5.4: |
Time-Dependent Vector Fields / 5.5: |
The Lie Derivative / 5.6: |
Invariant Tensor Fields / 5.7: |
Lie Groups / 5.8: |
Group Actions / 5.9: |
"One-Parameter Subgroups / 5.10: |
Left-and Right-Invariant Vector Fields on a Lie Group / 5.11: |
The Lie Algebra of a Lie Group / 5.12: |
Down-to-Earth Considerations / 5.13: |
The Adjoint Representation / 5.14: |
Integration and Fluxes / 6: |
Integration of Forms in Affine Spaces / 6.1: |
Integration of Forms on Chains in Manifolds / 6.2: |
Integration of Forms on Oriented Manifolds / 6.3: |
Fluxes in Continuum Physics / 6.4: |
General Bodies and Whitney's Geometric Integration Theory / 6.5: |
Further Topics / Part 3: |
Fibre Bundles / 7: |
Product Bundles / 7.1: |
Trivial Bundles / 7.2: |
General Fibre Bundles / 7.3: |
The Fundamental Existence Theorem / 7.4: |
The Tangent and Cotangent Bundles / 7.5: |
The Bundle of Linear Frames / 7.6: |
Principal Bundles / 7.7: |
Associated Bundles / 7.8: |
Fibre-Bundle Morphisms / 7.9: |
Cross Sections / 7.10: |
Iterated Fibre Bundles / 7.11: |
Inhomogeneity Theory / 8: |
Material Uniformity / 8.1: |
The Material Lie groupoid / 8.2: |
The Material Principal Bundle / 8.3: |
Flatness and Homogeneity / 8.4: |
Distributions and the Theorem of Frobenius / 8.5: |
JetBundles-and -Differential Equations / 8.6: |
Connection, Curvature, Torsion / 9: |
Ehresmann Connection / 9.1: |
Connections in Principal Bundles / 9.2: |
Linear Connections / 9.3: |
G-Connections / 9.4: |
Riemannian Connections / 9.5: |
Material Homogeneity / 9.6: |
Homogeneity Criteria / 9.7: |
A Primer in Continuum Mechanics / Appendix A: |
Bodies and Configurations / A.1: |
Observers and Frames / A.2: |
Strain / A.3: |
Volume and Area / A.4: |
The Material Time Derivative / A.5: |
Change of Reference / A.6: |
Transport Theorems / A.7: |
The General Balance Equation / A.8: |
The Fundamental Balance Equations of Continuum Mechanics / A.9: |
A Modicum of Constitutive Theory / A.10: |
Index |
Preface |
Motivation and Background / Part 1: |
The Case for Differential Geometry / 1: |
Classical Space-Time and Fibre Bundles / 1.1: |
Configuration Manifolds and Their Tangent and Cotangent Spaces / 1.2: |
The Infinite-dimensional Case / 1.3: |