Tensors have numerous applications in physics and engineering. There is often a fuzzy haze surrounding the concept of tensor that puzzles many students. The old-fashioned definition is difficult to understand because it is not rigorous; the modern definitions are difficult to understand because they are rigorous but at a cost of being more abstract and less intuitive. The goal of this book is to elucidate the concepts in an intuitive way but without loss of rigor, to help students gain deeper understanding. As a result, they will not need to recite those definitions in a parrot-like manner any more. This volume answers common questions and corrects many misconceptions about tensors. A large number of illuminating illustrations helps the reader to understand the concepts more easily. This unique reference text will benefit researchers, professionals, academics, graduate students and undergraduate students.
Author(s): Hongyu Guo
Publisher: World Scientific Publishing
Year: 2021
Language: English
Pages: 245
City: Singapore
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
