What Are Tensors Exactly?

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Contents Preface List of Boxes List of Figures Chapter Dependency Chart Notation Chapter 1. Confusions: What Are Tensors Exactly? §1. Questions and Confusions §2. Who Invented the Tensor? §3. Different Definitions of the Tensor §4. Plain Things by Fancy Tensor Names §5. Tensors without a Tensor Name—Linear Transformations §6. Comparison: Different Definitions of the Vector—Concrete Systems vs. Abstract Systems §7. Tensor Product and Tensor Spaces §8. Degree, Rank, Order or Dimension—Which Is the Best Name? *§9. What Are Pseudo-Scalars, Pseudo-Vectors and Pseudo-Tensors Exactly? §10. What Is Tensor Analysis Exactly? Relation to Riemannian Geometry 10.1 Vector Analysis 10.2 Tensor Analysis and Riemannian Geometry Chapter 2. Why and How Are Tensors Used in Machine Learning? §1. How AlphaGo Beat the Best Human Go Player via Deep Learning §2. The Tensor Data Structure 2.1 AlphaGo 2.2 Images and Videos 2.3 Speech and Audio Applications §3. TensorFlow and the Tensor Processing Unit (TPU) §4. Is Tensor in Machine Learning a Hype? Chapter 3. Direct Sum Space U ⊕ V §1. The Elements §2. The Operations §3. The Dimension of U ⊕ V Chapter 4. Gibbs Dyadics §1. What Is a Dyad? §2. When Are Two Dyads Equal? §3. What Are the Operations on Dyads? §4. What Is a Dyadic? §5. What Are the Operations on Dyadics? §6. When Are Two Dyadics Equal? §7. Matrix Representation §8. Change of Coordinates §9. What Are the Meanings of Dyadics? Linear Transformations and Bilinear Forms §10. What Is the Nature of Dyadic Juxtaposition? Chapter 5. Tensor Spaces (Tensor Product U ⊕ V) §1. Bilinear Mappings §2. Differences: Bilinear Mapping vs. Linear Mapping §3. Multilinear Mappings §4. Tensor Product Space of Two Vector Spaces §5. Decomposable Tensors §6. Tensor Product of Linear Mappings §7. Tensor Product Space of Multiple Vector Spaces §8. Vector-valued Tensors—The Most General Model Chapter 6. Tensor Spaces (Tensor Power V⊕(p;q)) §1. Tensor Spaces (Tensor Power Spaces) §2. Change of Basis §3. Induced Inner Product §4. Lowering and Raising Indices—Isomorphisms Chapter 7. Tensor Algebra §1. Tensor Product of Tensors §2. Tensor Algebra §3. Contraction of Tensors Chapter 8. Dynamics: The Inertia Tensor §1. Angular Momentum §2. Rotation of Rigid Body around a Fixed Point §3. Rotation of Rigid Body around a Fixed Axis §4. Parallel Axis Theorem and Perpendicular Axis Theorem §5. Ellipsoid of a Tensor Chapter 9. Electrodynamics: The EM Field Tensor §1. Electrodynamics in Tensor Formulation §2. Electrodynamics under Galilean Transformation 2.1 EM Field in the Form of Contravariant Tensor Fμν 2.2 EM Field in the Form of Covariant Tensor Fμν 2.3 EM Field in the Form of a Mixture of Fμν and Fμν 2.4 EM Field in the Form of a Mixture of Fμν and Fμν §3. Electrodynamics in Rotating Reference Frames *§4. Maxwell Equations in Exterior Differential Forms *§5. Proposal of New Notation d^ for Exterior Derivative Chapter 10. Riemannian Geometry and General Relativity §1. What Is "Curved Space" Exactly? 1.1 Extrinsic View of Curved Surfaces and Curved Spaces 1.2 Intrinsic View of Curved Surfaces due to Gauss 1.3 Riemann's Generalization of the Intrinsic Geometry §2. What Is a Tangent Space Exactly? 2.1 Extrinsic View Is Easy 2.2 Intrinsic View Is More Difficult §3. Tensor Transformation Laws Revisited §4. What Are the Differences? Differentiable Manifold vs. Riemannian Manifold §5. How Can Riemannian Geometry Be Applied to the Real World? —Conventionalism §6. What Is General Relativity Exactly? §7. What Is Time Exactly? Appendix 1. Topics of Linear Algebra §1. Proof of Commutativity of Addition §2. Covectors and the Dual Space §3. Inner Product §4. Contravariant and Covariant Components of Vectors 4.1 Contravariant coordinates as the parallel projections 4.2 Covariant coordinates as the perpendicular projections §5. Bilinear Forms and Quadratic Forms §6. Free Vector Spaces and Free Algebras 6.1 Intuitive Idea 6.2 Formal Definition of Free Vector Space 6.3 Free Algebras Appendix 2. Mathematical Structures §1. Mathematical Structures §2. Discrete Structures 2.1 Algebraic Structures 2.2 Order Structures §3. Continuous Structures 3.1 Topological Structures 3.2 Measure Structures §4. Mixed Structures Appendix 3. Axiomatic Systems §1. Undefined Concepts and Axioms §2. Axiomatic Systems—From Ancient to Modern Times §3. Consistency, Independence and Completeness 3.1 Consistency 3.2 Independence 3.3 Completeness Bibliography Index