دانلود کتاب Linear Algebra and Group Theory for Physicists and Engineers

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Preface How to Use the Book in Academic Courses? Roadmaps: How to Read the Book? Contents Part I Introduction to Linear Algebra 1 Vectors and Matrices 1.1 Vectors in Two and Three Dimensions 1.1.1 Two-Dimensional Vectors 1.1.2 Adding Vectors 1.1.3 Scalar Times Vector 1.1.4 Three-Dimensional Vectors 1.2 Vectors in Higher Dimensions 1.2.1 Multidimensional Vectors 1.2.2 Associative Law 1.2.3 The Origin 1.2.4 Multiplication and Its Laws 1.2.5 Distributive Laws 1.3 Complex Numbers and Vectors 1.3.1 Complex Numbers 1.3.2 Complex Vectors 1.4 Rectangular Matrix 1.4.1 Matrices 1.4.2 Adding Matrices 1.4.3 Scalar Times Matrix 1.4.4 Matrix Times Vector 1.4.5 Matrix-Times-Matrix 1.4.6 Distributive and Associative Laws 1.4.7 The Transpose Matrix 1.5 Square Matrix 1.5.1 Symmetric Square Matrix 1.5.2 The Identity Matrix 1.5.3 The Inverse Matrix as a Mapping 1.5.4 Inverse and Transpose 1.6 Complex Matrix and Its Hermitian Adjoint 1.6.1 The Hermitian Adjoint 1.6.2 Hermitian (Self-Adjoint) Matrix 1.7 Inner Product and Norm 1.7.1 Inner (Scalar) Product 1.7.2 Bilinearity 1.7.3 Skew-Symmetry 1.7.4 Norm 1.7.5 Normalization 1.7.6 Other Norms 1.7.7 Inner Product and the Hermitian Adjoint 1.7.8 Inner Product and a Hermitian Matrix 1.8 Orthogonal and Unitary Matrix 1.8.1 Inner Product of Column Vectors 1.8.2 Orthogonal and Orthonormal Column Vectors 1.8.3 Projection Matrix and Its Null Space 1.8.4 Unitary and Orthogonal Matrix 1.9 Eigenvalues and Eigenvectors 1.9.1 Eigenvectors and Their Eigenvalues 1.9.2 Singular Matrix and Its Null Space 1.9.3 Eigenvalues of the Hermitian Adjoint 1.9.4 Eigenvalues of a Hermitian Matrix 1.9.5 Eigenvectors of a Hermitian Matrix 1.10 The Sine Transform 1.10.1 Discrete Sine Waves 1.10.2 Orthogonality of the Discrete Sine Waves 1.10.3 The Sine Transform 1.10.4 Diagonalization 1.10.5 Sine Decomposition 1.10.6 Multiscale Decomposition 1.11 The Cosine Transform 1.11.1 Discrete Cosine Waves 1.11.2 Orthogonality of the Discrete Cosine Waves 1.11.3 The Cosine Transform 1.11.4 Diagonalization 1.11.5 Cosine Decomposition 1.12 Positive (Semi)definite Matrix 1.12.1 Positive Semidefinite Matrix 1.12.2 Positive Definite Matrix 1.13 Exercises: Generalized Eigenvalues 1.13.1 The Cauchy–Schwarz Inequality 1.13.2 The Triangle Inequality 1.13.3 Generalized Eigenvalues 1.13.4 Root of Unity and Fourier Transform 2 Determinant and Vector Product and Their Applications in Geometrical Mechanics 2.1 The Determinant 2.1.1 Minors and the Determinant 2.1.2 Examples 2.1.3 Algebraic Properties 2.1.4 The Inverse Matrix in Its Explicit Form 2.1.5 Cramer's Rule 2.2 Vector (Cross) Product 2.2.1 Standard Unit Vectors in 3-D 2.2.2 Inner Product—Orthogonal Projection 2.2.3 Vector (Cross) Product 2.2.4 The Right-Hand Rule 2.3 Orthogonalization 2.3.1 Invariance Under Orthogonal Transformation 2.3.2 Relative Axis System: Gram–Schmidt Process 2.3.3 Angle Between Vectors 2.4 Linear and Angular Momentum 2.4.1 Linear Momentum 2.4.2 Radial Component: Orthogonal Projection 2.4.3 Angular Momentum 2.4.4 Angular Momentum and Its Norm 2.4.5 Linear Momentum and Its Nonradial Component 2.4.6 Linear Momentum and Its Orthogonal Decomposition 2.5 Angular Velocity 2.5.1 Angular Velocity 2.5.2 The Rotating Axis System 2.5.3 Velocity and Its Decomposition 2.6 Real and Fictitious Forces 2.6.1 The Centrifugal Force 2.6.2 The Centripetal Force 2.6.3 The Euler Force 2.6.4 The Earth and Its Rotation 2.6.5 Coriolis Force 2.7 Exercises: Inertia and Principal Axes 2.7.1 Rotation and Euler Angles 2.7.2 Algebraic Right-Hand Rule 2.7.3 Linear Momentum and Its Conservation 2.7.4 Principal Axes 2.7.5 The Inertia Matrix 2.7.6 The Triple Vector Product 2.7.7 Linear Momentum: Orthogonal Decomposition 2.7.8 The Centrifugal and Centripetal Forces 2.7.9 The Inertia Matrix Times the Angular Velocity 2.7.10 Angular Momentum and Its Conservation 2.7.11 Rigid Body 2.7.12 The Percussion Point 2.7.13 Bohr's Atom and Energy Levels 3 Markov Matrix and Its Spectrum: Toward Search Engines 3.1 Characteristic Polynomial and Spectrum 3.1.1 Null Space and Characteristic Polynomial 3.1.2 Spectrum and Spectral Radius 3.2 Graph and Its Matrix 3.2.1 Weighted Graph 3.2.2 Markov Matrix 3.2.3 Example: Uniform Probability 3.3 Flow and Mass 3.3.1 Stochastic Flow: From State to State 3.3.2 Mass Conservation 3.4 The Steady State 3.4.1 The Spectrum of Markov Matrix 3.4.2 Converging Markov Chain 3.4.3 The Steady State 3.4.4 Search Engine in the Internet 3.5 Exercises: Gersgorin's Theorem 3.5.1 Gersgorin's Theorem 4 Special Relativity: Algebraic Point of View 4.1 Adding Velocities (or Speeds) 4.1.1 How to Add Velocities? 4.1.2 Einstein's Law: Never Exceed the Speed of Light! 4.1.3 Particle as Fast as Light 4.1.4 Singularity: Indistinguishable Particles 4.2 Systems and Their Time 4.2.1 Inertial Reference Frame 4.2.2 How to Measure Time? 4.2.3 The Self-system 4.2.4 Synchronization 4.3 Lorentz Group of Transformations (Matrices) 4.3.1 Space and Time: Same Status 4.3.2 Lorentz Transformation 4.3.3 Lorentz Matrix and the Infinity Point 4.3.4 Interchanging Coordinates 4.3.5 Composite Transformation 4.3.6 The Inverse Transformation 4.3.7 Abelian Group of Lorentz Matrices 4.4 Proper Time in the Self-system 4.4.1 Proper Time: Invariant 4.4.2 Time Dilation 4.4.3 Length Contraction 4.4.4 Simultaneous Events 4.5 Spacetime and Velocity 4.5.1 Doppler's Effect 4.5.2 Velocity in Spacetime 4.5.3 Moebius Transformation 4.5.4 Perpendicular Velocity 4.6 Relativistic Momentum and its Conservation 4.6.1 Invariant Mass 4.6.2 Momentum: Old Definition 4.6.3 Relativistic Momentum 4.6.4 Rest Mass vs. Relativistic Mass 4.6.5 Moderate (Nonrelativistic) Velocity 4.6.6 Closed System: Lose Mass—Gain Motion 4.6.7 The Momentum Matrix 4.6.8 Momentum and its Conservation 4.7 Relativistic Energy and its Conservation 4.7.1 Force: Derivative of Momentum 4.7.2 Open System: Constant Mass 4.7.3 Relativistic Energy: Kinetic Plus Potential 4.7.4 Moderate (Nonrelativistic) Velocity 4.8 Mass and Energy: Closed vs. Open System 4.8.1 Why Is It Called Rest Mass? 4.8.2 Mass is Invariant 4.8.3 Energy is Conserved—Mass Is Not 4.8.4 Particle Starting to Move 4.8.5 Say Mass, Not Rest Mass 4.8.6 Decreasing Mass in the Lab 4.8.7 Closed System: Energy Can Only Convert 4.8.8 Open System 4.8.9 Mass in a Closed System 4.9 Momentum–Energy and Their Transformation 4.9.1 New Mass 4.9.2 Spacetime 4.9.3 A Naive Approach 4.9.4 The Momentum–Energy Vector 4.9.5 The Momentum Matrix in Spacetime 4.9.6 Lorentz Transformation on Momentum–Energy 4.10 Energy and Mass 4.10.1 Invariant Nuclear Energy 4.10.2 Invariant Mass 4.10.3 Einstein's Formula 4.11 Center of Mass 4.11.1 Collection of Subparticles 4.11.2 Center of Mass 4.11.3 The Mass of the Collection 4.12 Oblique Force and Momentum 4.12.1 Oblique Momentum in x'-y' 4.12.2 View from Spacetime 4.12.3 The Lab: The New Self-system 4.13 Force in an Open System 4.13.1 Force in an Open Passive System 4.13.2 What Is the Force in Spacetime? 4.13.3 Proper Time in the Lab 4.13.4 Nearly Proper Time in the Lab 4.14 Perpendicular Force 4.14.1 Force: Time Derivative of Momentum 4.14.2 Passive System—Strong Perpendicular Force 4.15 Nonperpendicular Force 4.15.1 Force: Time Derivative of Momentum 4.15.2 Energy in an Open System 4.15.3 Open System—Constant Mass 4.15.4 Nearly Constant Energy in the Lab 4.15.5 Nonperpendicular Force: Same at All Systems 4.15.6 The Photon Paradox 4.16 Exercises: Special Relativity in 3-D 4.16.1 Lorentz Matrix and its Determinant 4.16.2 Motion in 3-D Part II Introduction to Group Theory 5 Groups and Isomorphism Theorems 5.1 Moebius Transformation and Matrix 5.1.1 Riemann Sphere—Extended Complex Plane 5.1.2 Moebius Transformation and the Infinity Point 5.1.3 The Inverse Transformation 5.1.4 Moebius Transformation as a Matrix 5.1.5 Product of Moebius Transformations 5.2 Matrix: A Function 5.2.1 Matrix as a Vector Function 5.2.2 Matrix Multiplication as Composition 5.3 Group and its Properties 5.3.1 Group 5.3.2 The Unit Element 5.3.3 Inverse Element 5.4 Mapping and Homomorphism 5.4.1 Mapping and its Origin 5.4.2 Homomorphism 5.4.3 Mapping the Unit Element 5.4.4 Preserving the Inverse Operation 5.4.5 Kernel of a Mapping 5.5 The Center and Kernel Subgroups 5.5.1 Subgroup 5.5.2 The Center Subgroup 5.5.3 The Kernel Subgroup 5.6 Equivalence Classes 5.6.1 Equivalence Relation in a Set 5.6.2 Decomposition into Equivalence Classes 5.6.3 Family of Equivalence Classes 5.6.4 Equivalence Relation Induced by a Subgroup 5.6.5 Equivalence Classes Induced by a Subgroup 5.7 The Factor Group 5.7.1 The New Set G/S 5.7.2 Normal Subgroup 5.7.3 The Factor (Quotient) Group 5.7.4 Is the Kernel Normal? 5.7.5 Isomorphism on the Factor Group 5.7.6 The Fundamental Theorem of Homomorphism 5.8 Geometrical Applications 5.8.1 Application in Moebius Transformations 5.8.2 Two-Dimensional Vector Set 5.8.3 Geometrical Decomposition into Planes 5.8.4 Family of Planes 5.8.5 Action of Factor Group 5.8.6 Composition of Functions 5.8.7 Oblique Projection: Extended Cotangent 5.8.8 Homomorphism onto Moebius Transformations 5.8.9 The Kernel 5.8.10 Eigenvectors and Fixed Points 5.8.11 Isomorphism onto Moebius Transformations 5.9 Application in Continued Fractions 5.9.1 Continued Fractions 5.9.2 Algebraic Formulation 5.9.3 The Approximants 5.9.4 Algebraic Convergence 5.10 Isomorphism Theorems 5.10.1 The Second Isomorphism Theorem 5.10.2 The Third Isomorphism Theorem 5.11 Exercises 6 Projective Geometry with Applications in Computer Graphics 6.1 Circles and Spheres 6.1.1 Degenerate ``Circle'' 6.1.2 Antipodal Points in the Unit Circle 6.1.3 More Circles 6.1.4 Antipodal Points in the Unit Sphere 6.1.5 General Multidimensional Hypersphere 6.1.6 Complex Coordinates 6.2 The Complex Projective Plane 6.2.1 The Complex Projective Plane 6.2.2 Topological Homeomorphism onto the Sphere 6.2.3 The Center and its Subgroups 6.2.4 Group Product 6.2.5 The Center—a Group Product 6.2.6 How to Divide by a Product? 6.2.7 How to Divide by a Circle? 6.2.8 Second and Third Isomorphism Theorems 6.3 The Real Projective Line 6.3.1 The Real Projective Line 6.3.2 The Divided Circle 6.4 The Real Projective Plane 6.4.1 The Real Projective Plane 6.4.2 Oblique Projection 6.4.3 Radial Projection 6.4.4 The Divided Sphere 6.4.5 Infinity Points 6.4.6 The Infinity Circle 6.4.7 Lines as Level Sets 6.5 Infinity Points and Line 6.5.1 Infinity Points and their Projection 6.5.2 Riemannian Geometry 6.5.3 A Joint Infinity Point 6.5.4 Two Lines Share a Unique Point 6.5.5 Parallel Lines Do Meet 6.5.6 The Infinity Line 6.5.7 Duality: Two Points Make a Unique Line 6.6 Conics and Envelopes 6.6.1 Conic as a Level Set 6.6.2 New Axis System 6.6.3 The Projected Conic 6.6.4 Ellipse, Hyperbola, or Parabola 6.6.5 Tangent Planes 6.6.6 Envelope 6.6.7 The Inverse Mapping 6.7 Duality: Conic–Envelope 6.7.1 Conic and its Envelope 6.7.2 Hyperboloid and its Projection 6.7.3 Projective Mappings 6.8 Applications in Computer Graphics 6.8.1 Translation 6.8.2 Motion in a Curved Trajectory 6.8.3 The Translation Matrix 6.8.4 General Translation of a Planar Object 6.8.5 Unavailable Tangent 6.8.6 Rotation 6.8.7 Relation to the Complex Projective Plane 6.9 The Real Projective Space 6.9.1 The Real Projective Space 6.9.2 Oblique Projection 6.9.3 Radial Projection 6.10 Duality: Point–Plane 6.10.1 Points and Planes 6.10.2 The Extended Vector Product 6.10.3 Three Points Make a Unique Plane 6.10.4 Three Planes Share a Unique Point 6.11 Exercises 7 Quantum Mechanics: Algebraic Point of View 7.1 Nondeterminism 7.1.1 Relativistic Observation 7.1.2 Determinism 7.1.3 Nondeterminism and Observables 7.2 State: Wave Function 7.2.1 Physical State 7.2.2 The Diagonal Position Matrix 7.2.3 Normalization 7.2.4 State and Its Overall Phase 7.2.5 Dynamics: Schrodinger Picture 7.2.6 Wave Function and Phase 7.2.7 Phase and Interference 7.3 Observables: Which Is First? 7.3.1 Measurement: The State Is Gone 7.3.2 The Momentum Matrix and Its Eigenvalues 7.3.3 Ordering Matters! 7.3.4 Commutator 7.3.5 Planck Constant 7.4 Observable and Its Expectation 7.4.1 Observable (Measurable) 7.4.2 Hermitian and Anti-Hermitian Parts 7.4.3 Symmetrization 7.4.4 Observation 7.4.5 Random Variable 7.4.6 Observable and Its Expectation 7.5 Heisenberg's Uncertainty Principle 7.5.1 Variance 7.5.2 Covariance 7.5.3 Heisenberg's Uncertainty Principle 7.6 Wave: Debroglie Relation 7.6.1 Infinite Matrix (or Operator) 7.6.2 Momentum: Another Operator 7.6.3 The Commutator 7.6.4 Wave: An Eigenfunction 7.6.5 Duality: Particle—Matter or Wave? 7.6.6 Debroglie's Relation: Momentum–Wave Number 7.7 Planck and Schrodinger Equations 7.7.1 Hamiltonian: Energy Operator 7.7.2 Time–Energy Uncertainty 7.7.3 Planck Relation: Frequency–Energy 7.7.4 No Potential: Momentum Is Conserved Too 7.7.5 Stability in Bohr's Atom 7.8 Eigenvalues 7.8.1 Shifting an Eigenvalue 7.8.2 Shifting an Eigenvalue of a Product 7.8.3 A Number Operator 7.8.4 Eigenvalue—Expectation 7.8.5 Down the Ladder 7.8.6 Null Space 7.8.7 Up the Ladder 7.9 Hamiltonian 7.9.1 Harmonic Oscillator 7.9.2 Concrete Number Operator 7.9.3 Energy Levels 7.9.4 Ground State (Zero-Point Energy) 7.9.5 Gaussian Distribution 7.10 Coherent State 7.10.1 Energy Levels and Their Superposition 7.10.2 Energy Levels and Their Precession 7.10.3 Coherent State 7.10.4 Probability to Have Certain Energy 7.10.5 Poisson Distribution 7.10.6 Conservation of Energy 7.11 Particle in 3-D 7.11.1 The Discrete 2-D Grid 7.11.2 Position and Momentum 7.11.3 Tensor Product 7.11.4 Commutativity 7.11.5 3-D Grid 7.11.6 Bigger Tensor Product 7.12 Angular Momentum 7.12.1 Angular Momentum Component 7.12.2 Using the Commutator 7.12.3 Up the Ladder 7.12.4 Down the Ladder 7.12.5 Angular Momentum 7.13 Toward the Path Integral 7.13.1 What Is an Electron? 7.13.2 Dynamics 7.13.3 Reversibility 7.13.4 Toward Spin 7.14 Exercises: Spin 7.14.1 Eigenvalues and Eigenvectors 7.14.2 Hamiltonian and Energy Levels 7.14.3 The Ground State and Its Conservation 7.14.4 Coherent State and Its Dynamics 7.14.5 Entanglement 7.14.6 Angular Momentum and Its Eigenvalues 7.14.7 Spin-One 7.14.8 Spin-One-Half and Pauli Matrices 7.14.9 Polarization 7.14.10 Conjugation 7.14.11 Dirac Matrices Anti-commute 7.14.12 Dirac Matrices in Particle Physics Part III Polynomials and Basis Functions 8 Polynomials and Their Gradient 8.1 Polynomials and Their Arithmetic Operations 8.1.1 Polynomial of One Variable 8.1.2 Real vs. Complex Polynomial 8.1.3 Addition 8.1.4 Scalar Multiplication 8.1.5 Multiplying Polynomials: Convolution 8.1.6 Example: Scalar Multiplication 8.2 Polynomial and Its Value 8.2.1 Value at a Given Point 8.2.2 The Naive Method 8.2.3 Using the Distributive Law 8.2.4 Recursion: Horner's Algorithm 8.2.5 Complexity: Mathematical Induction 8.3 Composition 8.3.1 Mathematical Induction 8.3.2 The Induction Step 8.3.3 Recursion: A New Horner Algorithm 8.4 Natural Number as a Polynomial 8.4.1 Decimal Polynomial 8.4.2 Binary Polynomial 8.5 Monomial and Its Value 8.5.1 Monomial 8.5.2 A Naive Method 8.5.3 Horner Algorithm: Implicit Form 8.5.4 Mathematical Induction 8.5.5 The Induction Step 8.5.6 Complexity: Total Cost 8.5.7 Recursion Formula 8.6 Differentiation 8.6.1 Derivative of a Polynomial 8.6.2 Second Derivative 8.6.3 High-Order Derivatives 8.7 Integration 8.7.1 Indefinite Integral 8.7.2 Definite Integral over an Interval 8.7.3 Examples 8.7.4 Definite Integral over the Unit Interval 8.8 Sparse Polynomials 8.8.1 Sparse Polynomial 8.8.2 Sparse Polynomial: Explicit Form 8.8.3 Sparse Polynomial: Recursive Form 8.8.4 Improved Horner Algorithm 8.8.5 Power of a Polynomial 8.8.6 Composition 8.9 Polynomial of Two Variables 8.9.1 Polynomial of Two Independent Variables 8.9.2 Arithmetic Operations 8.10 Differentiation and Integration 8.10.1 Partial Derivatives 8.10.2 The Gradient 8.10.3 Integral over the Unit Triangle 8.10.4 Second Partial Derivatives 8.10.5 Degree 8.11 Polynomial of Three Variables 8.11.1 Polynomial of Three Independent Variables 8.12 Differentiation and Integration 8.12.1 Partial Derivatives 8.12.2 The Gradient 8.12.3 Vector Field (or Function) 8.12.4 The Jacobian 8.12.5 Integral over the Unit Tetrahedron 8.13 Normal and Tangential Derivatives 8.13.1 Directional Derivative 8.13.2 Normal Derivative 8.13.3 Differential Operator 8.13.4 High-Order Normal Derivatives 8.13.5 Tangential Derivative 8.14 High-Order Partial Derivatives 8.14.1 High-Order Partial Derivatives 8.14.2 The Hessian 8.14.3 Degree 8.15 Exercises: Convolution 8.15.1 Convolution and Polynomials 8.15.2 Polar Decomposition 9 Basis Functions: Barycentric Coordinates in 3-D 9.1 Tetrahedron and its Mapping 9.1.1 General Tetrahedron 9.1.2 Integral Over a Tetrahedron 9.1.3 The Chain Rule 9.1.4 Degrees of Freedom 9.2 Barycentric Coordinates in 3-D 9.2.1 Barycentric Coordinates in 3-D 9.2.2 The Inverse Mapping 9.2.3 Geometrical Interpretation 9.2.4 The Chain Rule and Leibniz Rule 9.2.5 Integration in Barycentric Coordinates 9.3 Independent Degrees of Freedom 9.3.1 Continuity Across an Edge 9.3.2 Smoothness Across an Edge 9.3.3 Continuity Across a Side 9.3.4 Independent Degrees of Freedom 9.4 Piecewise-Polynomial Functions 9.4.1 Smooth Piecewise-Polynomial Function 9.4.2 Continuous Piecewise-Polynomial Function 9.5 Basis Functions 9.5.1 Side Midpoint Basis Function 9.5.2 Edge-Midpoint Basis Function 9.5.3 Hessian-Related Corner Basis Function 9.5.4 Gradient-Related Corner Basis Function 9.5.5 Corner Basis Function 9.6 Numerical Experiment: Electromagnetic Waves 9.6.1 Frequency and Wave Number 9.6.2 Adaptive Mesh Refinement 9.7 Numerical Results 9.7.1 High-Order Finite Elements 9.7.2 Linear Adaptive Finite Elements 9.8 Exercises Part IV Finite Elements in 3-D 10 Automatic Mesh Generation 10.1 The Refinement Step 10.1.1 Iterative Multilevel Refinement 10.1.2 Conformity 10.1.3 Regular Mesh 10.1.4 How to Preserve Regularity? 10.2 Approximating a 3-D Domain 10.2.1 Implicit Domain 10.2.2 Example: A Nonconvex Domain 10.2.3 How to Find a Boundary Point? 10.3 Approximating a Convex Boundary 10.3.1 Boundary Refinement 10.3.2 Boundary Edge and Triangle 10.3.3 How to Fill a Valley? 10.3.4 How to Find a Boundary Edge? 10.3.5 Locally Convex Boundary: Gram–Schmidt Process 10.4 Approximating a Nonconvex Domain 10.4.1 Locally Concave Boundary 10.4.2 Convex Meshes 10.5 Exercises 11 Mesh Regularity 11.1 Angle and Sine in 3-D 11.1.1 Sine in a Tetrahedron 11.1.2 Minimal Angle 11.1.3 Proportional Sine 11.1.4 Minimal Sine 11.2 Adequate Equivalence 11.2.1 Equivalent Regularity Estimates 11.2.2 Inadequate Equivalence 11.2.3 Ball Ratio 11.3 Numerical Experiment 11.3.1 Mesh Regularity 11.3.2 Numerical Results 11.4 Exercises 12 Numerical Integration 12.1 Integration in 3-D 12.1.1 Volume of a Tetrahedron 12.1.2 Integral in 3-D 12.1.3 Singularity 12.2 Changing Variables 12.2.1 Spherical Coordinates 12.2.2 Partial Derivatives 12.2.3 The Jacobian 12.2.4 Determinant of Jacobian 12.2.5 Integrating a Composite Function 12.3 Integration in the Meshes 12.3.1 Integrating in a Ball 12.3.2 Stopping Criterion 12.3.3 Richardson Extrapolation 12.4 Exercises 13 Spline: Variational Model in 3-D 13.1 Expansion in Basis Functions 13.1.1 Degrees of Freedom in the Mesh 13.1.2 The Function Space and Its Basis 13.2 The Stiffness Matrix 13.2.1 Assemble the Stiffness Matrix 13.2.2 How to Order the Basis Functions? 13.3 Finding the Optimal Spline 13.3.1 Minimum Energy 13.3.2 The Schur Complement 13.4 Exercises Part V Permutation Group and Determinant in Quantum Chemistry 14 Permutation Group and the Determinant 14.1 Permutation 14.1.1 Permutation 14.1.2 Switch 14.1.3 Cycle 14.2 Decomposition 14.2.1 Composition (Product) 14.2.2 3-Cycle 14.2.3 4-Cycle 14.2.4 How to Decompose a Permutation? 14.3 Permutations and Their Group 14.3.1 Group of Permutations 14.3.2 How Many Permutations? 14.4 Determinant 14.4.1 Determinant: A New Definition 14.4.2 Determinant of the Transpose 14.4.3 Determinant of a Product 14.4.4 Orthogonal Matrix 14.4.5 Unitary Matrix 14.5 The Characteristic Polynomial 14.5.1 The Characteristic Polynomial 14.5.2 Trace—Sum of Eigenvalues 14.5.3 Determinant—Product of Eigenvalues 14.6 Exercises: Permutation and Its Structure 14.6.1 Decompose as a Product of Switches 15 Electronic Structure in the Atom: The Hartree–Fock System 15.1 Wave Function 15.1.1 Particle and Its Wave Function 15.1.2 Entangled Particles 15.1.3 Disentangled Particles 15.2 Electrons in Their Orbitals 15.2.1 Atom: Electrons in Orbitals 15.2.2 Potential Energy and Its Expectation 15.3 Distinguishable Electrons 15.3.1 Hartree Product 15.3.2 Potential Energy of the Hartree Product 15.4 Indistinguishable Electrons 15.4.1 Indistinguishable Electrons 15.4.2 Pauli's Exclusion Principle: Slater Determinant 15.5 Orbitals and Their Canonical Form 15.5.1 The Overlap Matrix and Its Diagonal Form 15.5.2 Unitary Transformation 15.5.3 Orthogonal Orbitals 15.5.4 Slater Determinant and Unitary Transformation 15.5.5 Orthonormal Orbitals: The Canonical Form 15.5.6 Slater Determinant and Its Overlap 15.6 Expected Energy 15.6.1 Coulomb and Exchange Integrals 15.6.2 Effective Potential Energy 15.6.3 Kinetic Energy 15.6.4 The Schrodinger Equation in Its Integral Form 15.7 The Hartree–Fock System 15.7.1 Basis Functions and the Coefficient Matrix 15.7.2 The Mass Matrix 15.7.3 The Pseudo-Eigenvalue Problem 15.7.4 Is the Canonical Form Plausible? 15.8 Exercises: Electrostatic Potential 15.8.1 Potential: Divergence of Flux Part VI The Jordan Form 16 The Jordan Form 16.1 Nilpotent Matrix and Generalized Eigenvectors 16.1.1 Nilpotent Matrix 16.1.2 Cycle and Invariant Subspace 16.1.3 Generalized Eigenvectors and Their Linear Independence 16.1.4 More General Cases 16.1.5 Linear Dependence 16.1.6 More General Cases 16.2 Nilpotent Matrix and Its Jordan Form 16.2.1 How to Design a Jordan Basis? 16.2.2 The Reverse Ordering 16.2.3 Jordan Blocks 16.2.4 Jordan Blocks and Their Powers 16.3 General Matrix 16.3.1 Characteristic Polynomial: Eigenvalues and Their Multiplicity 16.3.2 Block and Its Invariant Subspace 16.3.3 Block and Its Jordan Form 16.3.4 Block and Its Characteristic Polynomial 16.4 Exercises: Hermitian Matrix and Its Eigenbasis 16.4.1 Nilpotent Hermitian Matrix 16.4.2 Hermitian Matrix and Its Orthonormal Eigenbasis 17 Jordan Decomposition of a Matrix 17.1 Greatest Common Divisor 17.1.1 Integer Division with Remainder 17.1.2 Congruence: Same Remainder 17.1.3 Common Divisor 17.1.4 The Euclidean Algorithm 17.1.5 The Extended Euclidean Algorithm 17.1.6 Confining the Coefficients 17.1.7 The Modular Extended Euclidean Algorithm 17.2 Modular Arithmetic 17.2.1 Coprime 17.2.2 Modular Multiplication 17.2.3 Modular Power 17.2.4 Modular Inverse 17.2.5 How to Find the Modular Inverse? 17.3 The Chinese Remainder Theorem 17.3.1 Modular Equation 17.3.2 How to Use the Coprime? 17.3.3 Modular System of Equations 17.3.4 Uniqueness 17.4 How to Use the Remainder? 17.4.1 Integer Decomposition 17.4.2 Binary Number 17.4.3 Horner's Algorithm 17.5 Polynomials and the Chinese Remainder Theorem 17.5.1 Characteristic Polynomial: Root and its Multiplicity 17.5.2 Multiplicity and Jordan Subspace 17.5.3 The Chinese Remainder Theorem with Polynomials 17.6 The Jordan Decomposition 17.6.1 Properties of Q 17.6.2 The Diagonal Part 17.6.3 The Nilpotent Part 17.6.4 The Jordan Decomposition 17.7 Example: Space of Polynomials 17.7.1 Polynomial and its Differentiation 17.7.2 The Jordan Basis 17.7.3 The Jordan Block 17.7.4 The Jordan Block and its Powers 17.8 Exercises: Numbers—Polynomials 17.8.1 Natural Numbers: Binary Form 18 Algebras and Their Derivations and Their Jordan Form 18.1 Eigenfunctions 18.1.1 Polynomials of Any Degree 18.1.2 Eigenfunction 18.1.3 The Leibniz (Product) Rule 18.1.4 Mathematical Induction 18.2 Algebra and Its Derivation 18.2.1 Leibniz Rule 18.2.2 Product (Multiplication) 18.2.3 Derivation 18.3 Product and Its Derivation 18.3.1 Two-Level (Virtual) Binary Tree 18.3.2 Multilevel Tree 18.3.3 Pascal's Triangle and the Binomial Formula 18.4 Product and Its Jordan Subspace 18.4.1 Two Members from Two Jordan Subspaces 18.4.2 Product and Its New Jordan Subspace 18.4.3 Example: Polynomials Times Exponents 18.5 Derivation on a Subalgebra 18.5.1 Restriction to a Subalgebra 18.6 More Examples 18.6.1 Smooth Functions 18.6.2 Finite Dimension 18.7 The Diagonal Part 18.7.1 Is It a Derivation? 18.8 Exercises: Derivation and Its Exponent 18.8.1 Exponent—Product Preserving Part VII Linearization in Numerical Relativity 19 Einstein Equations and their Linearization 19.1 How to Discretize and Linearize? 19.1.1 How to Discretize in Time? 19.1.2 How to Differentiate the Metric? 19.1.3 The Flat Minkowski Metric 19.1.4 Riemann's Normal coordinates 19.1.5 Toward Gravity Waves 19.1.6 Where to Linearize? 19.2 The Christoffel Symbol 19.2.1 The Gradient Symbol and its Variation 19.2.2 The Inverse Matrix and its Variation 19.2.3 Einstein Summation Convention 19.2.4 The Christoffel Symbol and its Variation 19.3 Einstein Equations in Vacuum 19.3.1 The Riemann Tensor and its Variation 19.3.2 Vacuum and Curvature 19.3.3 The Ricci Tensor 19.3.4 Einstein Equations in Vacuum 19.3.5 Stable Time Marching 19.4 The Trace-Subtracted Form 19.4.1 The Stress Tensor 19.4.2 The Stress Tensor and its Variation 19.4.3 The Trace-Subtracted Form 19.5 Einstein Equations: General Form 19.5.1 The Ricci Scalar 19.5.2 The Einstein Tensor 19.5.3 Einstein Equations—General Form 19.6 How to Integrate? 19.6.1 Integration by Parts 19.6.2 Why to Linearize? 19.6.3 Back to the Trace-Subtracted Form 19.7 Exercises References Index