Geometry for Physicists

Transcription

1 Hung Nguyen-Schafer Jan-Philip Schmidt Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers 4 i Springer

2 Contents 1 General Basis and Bra-Ket Notation Introduction to General Basis and Tensor Types General Basis in Curvilinear Coordinates Orthogonal Cylindrical Coordinates Orthogonal Spherical Coordinates Eigenvalue Problem of a Linear Coupled Oscillator Notation of Bra and Ket Properties of Kets Analysis of Bra and Ket Bra and Ket Bases Gram-Schmidt Scheme of Basis Orthonormalization Cauchy-Schwarz and Triangle Inequalities Computing Ket and Bra Components Inner Product of Two Kets Outer Product of Bra and Ket Ket and Bra Projection Components on the Bases Linear Transformation of Kets Coordinate Transformations Hermitian Transformation Applying Bra and Ket Analysis to Eigenvalue Problems 31 References 34 2 Tensor Analysis Introduction to Tensors Definition of Tensors Tensor Algebra General Bases in General Curvilinear Coordinates Metric Coefficients in General Curvilinear Coordinates Tensors of Second Order and Higher Orders Tensor and Cross Products of Two Vectors in General Bases Rules of Tensor Calculations 56 ix

3 x Contents 2.4 Coordinate Transformations Transformation in the Orthonormal Coordinates Transformation of Curvilinear Coordinates in EN Examples of Coordinate Transformations Transformation of Curvilinear Coordinates in RN Tensor Calculus in General Curvilinear Coordinates Physical Component of Tensors Derivatives of Covariant Bases Christoffel Symbols of First and Second Kind Prove that the Christoffel Symbols are... Symmetric Examples of Computing the Christoffel Symbols Coordinate Transformations of the Christoffel Symbols Derivatives of Contravariant Bases Derivatives of Covariant Metric Coefficients Covariant Derivatives of Tensors Riemann-Christoffel Tensor Ricci's Lemma Derivative of the Jacobian Ricci Tensor Einstein Tensor 97 References 99 3 Elementary Differential Geometry Introduction Arc Length and Surface in Curvilinear Coordinates Unit Tangent and Normal Vector to Surface The First Fundamental Form The Second Fundamental Form Gaussian and Mean Curvatures Ill 3.7 Riemann Curvature Gauss-Bonnet Theorem Gauss Derivative Equations Weingarten's Equations Gauss-Codazzi Equations Lie Derivatives Vector Fields in Riemannian Manifold Lie Bracket Lie Dragging Lie Derivatives Torsion and Curvature in a Distorted and Curved Manifold Killing Vector Fields 135

4 Contents xi 3.13 Invariant Time Derivatives on Moving Surfaces Invariant Time Derivative of an Invariant Field Invariant Time Derivative of Tensors 140 References Applications of Tensors and Differential Geometry Nabla Operator in Curvilinear Coordinates Gradient, Divergence, and Curl Gradient of an Invariant Gradient of a Vector Divergence of a Vector Divergence of a Second-Order Tensor Curl of a Covariant Vector Laplacian Operator Laplacian of an Invariant Laplacian of a Contravariant Vector Applying Nabla Operators in Spherical Coordinates Gradient of an Invariant Divergence of a Vector Curl of a Vector The Divergence Theorem Gauss and Stokes Theorems Green's Identities First Green's Identity Second Green's Identity Differentials of Area and Volume Calculating the Differential of Area Calculating the Differential of Volume Governing Equations of Computational Fluid Dynamics Continuity Equation Momentum Equations Energy (Rothalpy) Equation Basic Equations of Continuum Mechanics Cauchy's Law of Motion Principal Stresses of Cauchy's Stress Tensor Cauchy's Strain Tensor Constitutive Equations of Elasticity Laws Maxwell's Equations of Electrodynamics Einstein Field Equations Schwarzschild's Solution of the Einstein Field Equations Schwarzschild Black Hole 193 References 196

5 xii Contents Appendix A: Relations Between Covariant and Contravariant Bases 197 Appendix B: Physical Components of Tensors 203 Appendix C: Nabla Operators 207 Appendix D: Essential Tensors 211 Appendix E: Euclidean and Riemannian Manifolds 215 Definitions of Mathematical Symbols in this Book 237 Index 239