Intro
Preface
Contents
1 Introduction
2 Prelude: Finite-Dimensional Systems
2.1 The Framework for the Problem of Statics
2.2 On the Solutions of the Problem of Statics
2.2.1 Existence of Solutions
2.2.2 Static Indeterminacy and Optimal Solutions
2.2.3 Worst-Case Loading and Load Capacity
Part I Algebraic Theory: Uniform Fluxes
3 Simplices in Affine Spaces and Their Boundaries
3.1 Affine Spaces: Notation
3.2 Simplices
3.3 Cubes, Prisms, and Simplices
3.4 Orientation
3.5 Simplices on the Boundaries and Their Orientations
3.6 Subdivisions

4 Uniform Fluxes in Affine Spaces
4.1 Basic Assumptions
4.2 Balance and Linearity
4.3 Immediate Implications of Skew Symmetry and Multi-Linearity
4.4 The Algebraic Cauchy Theorem
5 From Uniform Fluxes to Exterior Algebra
5.1 Polyhedral Chains and Cochains
5.2 Component Representation of Fluxes
5.3 Multivectors
5.4 Component Representation of Multivectors
5.5 Alternation
5.6 Exterior Products
5.7 The Spaces of Multivectors and Multi-Covectors
5.8 Contraction
5.9 Pullback of Alternating Tensors
5.10 Abstract Algebraic Cauchy Formula

Part II Smooth Theory
6 Smooth Analysis on Manifolds: A Short Review
6.1 Manifolds and Bundles
6.1.1 Manifolds
6.1.2 Tangent Vectors and the Tangent Bundle
6.1.3 Fiber Bundles
6.1.4 Vector Bundles
6.1.5 Tangent Mappings
6.1.6 The Tangent Bundle of a Fiber Bundle and Its Vertical Subbundle
6.1.7 Jet Bundles
6.1.8 The First Jet of a Vector Bundle
6.1.9 The Pullback of a Fiber Bundle
6.1.10 Dual Vector Bundles and the Cotangent Bundle
6.2 Tensor Bundles and Differential Forms
6.2.1 Tensor Bundles and Their Sections
6.2.2 Differential Forms

6.2.3 Contraction and Related Mappings
6.2.4 Vector-Valued Forms
6.2.5 Density-Dual Spaces
6.3 Differentiation and Integration
6.3.1 Integral Curves and the Flow of a Vector Field
6.3.2 Exterior Derivatives
6.3.3 Partitions of Unity
6.3.4 Orientation on Manifolds
6.3.5 Integration on Oriented Manifolds
6.3.6 Stokes's Theorem
6.3.7 Integration Over Chains on Manifolds
6.4 Manifolds with Corners
7 Interlude: Smooth Distributions of Defects
7.1 Introduction
7.2 Forms and Hypersurfaces
7.3 Layering Forms, Defect Forms

7.4 Smooth Distributions of Dislocations
7.5 Inclinations and Disclinations, the Smooth Case
7.6 Frank's Rules for Smooth Distributions of Defects
8 Smooth Fluxes
8.1 Balance Principles and Fluxes
8.1.1 Densities of Extensive Properties
8.1.2 Flux Forms and Cauchy's Formula
8.1.3 Extensive Properties and Cauchy Formula-Local Representation
8.1.4 The Cauchy Flux Theorem
8.1.4.1 Assumptions
8.1.4.2 Notation
8.1.4.3 Construction
8.1.5 The Differential Balance Law
8.2 Properties of Fluxes
8.2.1 Flux Densities and Orientation