Intro; Preface; Contents; 1 General Introduction; 1.1 Classical Physics, Lagrangian, and Invariance; 1.2 General Covariance, Gauge Invariance; 1.3 Objectives and Planning; 2 Basic Concepts on Manifolds, Spacetimes, and Calculusof Variations; 2.1 Introduction; 2.2 Space-Time Background; 2.2.1 Basics on Flat Minkowski Spacetime; 2.2.2 Twisted and Curved Spacetimes; 2.3 Manifolds, Tensor Fields, and Connections; 2.3.1 Coordinate System, and Group of Transformations; 2.3.1.1 Manifolds, Tangent Space, Cotangent Space; 2.3.1.2 Change of Coordinate System

2.3.1.3 Examples of Group of Transformations2.3.1.4 Lorentz Invariance; 2.3.2 Elements on Spacetime and Invariance for Relativity; 2.3.2.1 Forms, Tensors and (Pseudo)-Riemannian Manifolds; 2.3.2.2 Hilbert's Causality Principle; 2.3.2.3 Euclidean Spacetime and Isometries; 2.3.2.4 Minkowski Spacetime and Lorentz Transformations; 2.3.2.5 Global Poincaré Transformations; 2.3.3 Volume-Form; 2.3.4 Affine Connection; 2.3.4.1 Affine Connection, Affinely Connected Manifold; 2.3.4.2 Example: Spherical Coordinate System; 2.3.4.3 Example: Elliptic-Hyperbolic Coordinate System

2.3.4.4 Practical Formula for Covariant Derivative2.3.4.5 Torsion and Curvature; 2.3.4.6 Newtonian Spacetime; 2.3.4.7 Levi-Civita Connection; 2.3.4.8 Normal Coordinate System and Inertial Frame; 2.3.5 Tetrads and Affine Connection: Continuum Transformations; 2.3.5.1 Transformation of a Continuum; 2.3.5.2 Holonomic Mapping; 2.3.5.3 Nonholonomic Mapping and Torsion e.g. (2000); 2.3.5.4 Nonholonomic Transformation and Curvature; 2.3.5.5 Torsion, Curvature, and Smoothness of Tensor Fields; 2.4 Invariance for Lagrangian and Euler-Lagrange Equations

2.4.1 Covariant Formulation of Classical Mechanics of a Particle2.4.2 Basic Elements for Calculus of Variations; 2.4.3 Extended Euler-Lagrange Equations; 2.5 Simple Examples in Continuum and Relativistic Mechanics; 2.5.1 Particles in a Minkowski Spacetime; 2.5.2 Some Continua Examples; 2.5.2.1 Energy-Momentum Tensor; 2.5.2.2 Dust in Relativistic Mechanics; 2.5.2.3 Perfect Fluids in Relativistic Mechanics; 2.5.2.4 Strain Gradient Continuum; 3 Covariance of Lagrangian Density Function; 3.1 Introduction; 3.2 Some Basic Theorems; 3.2.1 Theorem of Cartan; 3.2.2 Theorem of lovelockarma

3.2.3 Theorem of Quotient3.3 Invariance with Respect to the Metric; 3.3.1 Transformation Rules for the Metric and Its Derivatives; 3.3.2 Introduction of Dual Variables; 3.3.3 Theorem; 3.4 Invariance with Respect to the Connection; 3.4.1 Preliminary; 3.4.2 Application: Covariance of L; 3.4.3 Summary for Lagrangian Covariance; 3.4.4 Covariance of Nonlinear Elastic Continuum; 3.4.4.1 Covariance of Strain Energy Density; 3.4.4.2 Examples of Nonlinear Elastic Material Models; 4 Gauge Invariance for Gravitation and Gradient Continuum; 4.1 Introduction to Gauge Invariance