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Astronomical Distance Units .

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Celestial Coordinates .

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Celestial Navigation .

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Location of North and South Celestial Poles .

Chemistry

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Avogadro's Number .

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Balancing Chemical Equations

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The Periodic Table .

Classical Mechanics

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Blackbody Radiation .

Classical Physics

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Archimedes Principle

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Bernoulli Principle

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Center of Mass Frame

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Comparison Between Gravitation and Electrostatics

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Compton Effect .

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Cyclotron Resonance

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Double Slit Experiment

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Elastic and Inelastic Collisions .

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Gauss's Law of Universal Gravity .

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Cosmology

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Cosmic Background Radiation and Decoupling .

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Inflation Theory

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Introduction to Black Holes .

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Olbers' Paradox .

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Stephen Hawking's PhD Thesis .

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Finance and Accounting

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Game Theory

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General Relativity

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Catalog of Spacetimes .

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Curvature and Parallel Transport

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Einstein's Field Equations

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Gravitational Waves

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Hyperbolic Motion and Rindler Coordinates .

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Quantum Gravity

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Ricci Decomposition .

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Stress-Energy-Momentum Tensor .

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The Area Metric

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The Dirac Equation in Curved Spacetime .

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The Equivalence Principal

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The Essential Mathematics of General Relativity

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The Light Cone .

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The Metric Tensor .

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The Principle of Least Action in Relativity .

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Vierbein (Frame) Fields

Group Theory

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Basic Group Theory .

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Basic Representation Theory .

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Building Groups From Other Groups .

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Sets, Groups, Modules, Rings and Vector Spaces

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Symmetric Groups .

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The Integers Modulo n Under + and x .

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory .

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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics .

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Hamiltonian Formulation .

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Liouville's Theorem

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Mathematics

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Amplitude, Period and Phase

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Augmented Matrices and Cramer's Rule

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Binomial Theorem (Pascal's Triangle)

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Completing the Square

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Complex Numbers

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Composite Functions

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Conformal Transformations .

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Conjugate Pair Theorem

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Contravariant and Covariant Components of a Vector

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Derivatives of Inverse Functions

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Double Angle Formulas

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Eigenvectors and Eigenvalues

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Euler Formula for Polyhedrons

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Factoring of a³ +/- b³

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Fourier Series and Transforms .

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Gauss's Divergence Theorem

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Grassmann and Clifford Algebras .

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Heron's Formula

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Index Notation (Tensors and Matrices) .

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Integration By Parts

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Introduction to Conformal Field Theory .

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Inverse of a Function

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Law of Sines and Cosines

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Line Integrals, ∮

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Logarithms and Logarithmic Equations

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Matrices and Determinants

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Matrix Exponential

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Mean Value and Rolle's Theorem

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Modulus Equations

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Orthogonal Curvilinear Coordinates .

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Parabolas, Ellipses and Hyperbolas

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Piecewise Functions

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Polar Coordinates

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Polynomial Division

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Quaternions 1 .

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Quaternions 2 .

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Regular Polygons

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Similar Matrices and Diagonalization .

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Spherical Trigonometry

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Stirling's Approximation

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Sum and Differences of Squares and Cubes

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Tangent and Normal Line

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Taylor and Maclaurin Series .

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The Essential Mathematics of Lie Groups

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The Limit Definition of the Exponential Function

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Tic-Tac-Toe Factoring

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Trapezoidal Rule

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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Nuclear Physics

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Radioactive Decay

Particle Physics

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Feynman Diagrams and Loops

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Field Dimensions

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Helicity, Chirality and Weyl Spinors .

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Klein-Gordon and Dirac Equations .

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Regularization and Renormalization

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Scattering - Mandelstam Variables

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Spin 1 Eigenvectors .

Probability and Statistics

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Box and Whisker Plots

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Buffon's Needle .

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Categorical Data - Crosstabs

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Chebyshev's Theorem

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Chi Squared Goodness of Fit

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Conditional Probability

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Confidence Intervals

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Factor Analysis

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Hypothesis Testing

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Linear Regression

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Monte Carlo Methods

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Non Parametric Tests

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Pearson Correlation

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Permutations and Combinations

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Pooled Variance and Standard Error

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Probability Distributions

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Probability Rules

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Sample Size Determination

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Sampling Distributions

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Set Theory - Venn Diagrams

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Stacked and Unstacked Data

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Survey Data - Likert Item and Scale

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How this site works ...

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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol

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Public Key Encryption

Quantum Computing

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Density Operators and Mixed States .

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Entangled States .

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Quantum Field Theory

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Creation and Annihilation Operators

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Field Operators for Bosons and Fermions

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Lagrangians in Quantum Field Theory

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Path Integral Formulation

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Relativistic Quantum Field Theory

Quantum Mechanics

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Clebsch-Gordan Coefficients .

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Electron Orbital Angular Momentum and Spin

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Heisenberg Uncertainty Principle

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Ladder Operators .

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Multi Electron Wavefunctions .

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Pauli Spin Matrices

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Photoelectric Effect .

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Position and Momentum States .

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Probability Current

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Schrodinger Equation for Hydrogen Atom .

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Schrodinger Wave Equation

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Spin 1/2 Eigenvectors

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The Differential Operator

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The Essential Mathematics of Quantum Mechanics

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The Quantum Harmonic Oscillator .

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The Schrodinger, Heisenberg and Dirac Pictures

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The WKB Approximation

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Time Dependent Perturbation Theory

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Time Evolution and Symmetry Operations

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Time Independent Perturbation Theory

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Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .

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Fermi-Dirac Statistics .

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Intrinsic and Extrinsic Semiconductors .

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The P-N Junction

Special Relativity

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Electromagnetic (Faraday) Tensor .

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Energy and Momentum in Special Relativity, E = mc² .

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Invariance of the Velocity of Light .

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Lorentz Invariance .

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Lorentz Transform .

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Lorentz Transformation of the EM Field .

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Newton versus Einstein

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Spinors - Part 1 .

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Spinors - Part 2 .

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The Continuity Equation .

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The Lorentz Group .

Statistical Mechanics

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Entropy and the Partition Function

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The Harmonic Oscillator

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String Theory

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Bosonic Strings

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Extra Dimensions

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Introduction to String Theory

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Kaluza-Klein Compactification of Closed Strings

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Strings in Curved Spacetime

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Toroidal Compactification

Superconductivity

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Bardeen–Cooper–Schrieffer Theory

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Introduction to Superconductivity .

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Superconductivity (Lectures 1 - 10)

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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields

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Generators of a Supergroup

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Grassmann Numbers

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Introduction to Supersymmetry

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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)

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Gauge Theories (Yang-Mills)

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Gravitational Force and the Planck Scale

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Introduction to the Standard Model

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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge

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Quantum Flavordynamics and Quantum Chromodynamics

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Special Unitary Groups and the Standard Model - Part 1 .

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Special Unitary Groups and the Standard Model - Part 2

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Special Unitary Groups and the Standard Model - Part 3 .

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Standard Model Lagrangian

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The Higgs Mechanism

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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Last modified: June 13, 2022 ✓


Index Notation (Tensors and Matrices)
-------------------------------------

Problems With Matrices
----------------------

1.  They can be big and difficult to manipulate.

2.  They do not commute so the order of multiplication 
    is important.

3.  Some things cannot be represented by matrices or 
    matrix algebra.

Consider the 2D rotation matrix:

 -  -      -   -     -          - -   -   
| x¹ | -> | x^1' | = |  cosθ sinθ || x¹ |
| x² |    | x^2' |   | -sinθ cosθ || x² |
 -  -      -   -     -          - -   -  

                     -                -
                  = |  cosθx¹ + sinθx² |
                    | -sinθx¹ + cosθx² |
                     -                -

Rewrite the rotation matrix as:

 -         -
| Λ^1'₁  Λ^1'₂ |
| Λ^2'₁  Λ^2'₂ |
 -         -

xⁱ -> x^i' = ΣΛ^i'_jx^j  
          ^j

       = Λ^i'_jx^j using Einstein's notation.

       = Λ^i'₁x¹ + Λ^i'₂x²

So we get:

x^1'  = cosθx¹ + sinθx²

and,

x^2'  = -sinθx¹ + cosθx² 

In index notation the order doesn't matter.  Thus,

x^jΛ^i'_j ≡ Λ^i'_jx^j

However, this not true for matrices.

 -   -     -  -  -          -
| x^1' | = | x¹ ||  cosθ sinθ |
| x^2' |   | x² || -sinθ cosθ |
 -   -     -  -  -          -

which is an illegal operation.

Converting Index to Matrix Notation
-----------------------------------

To get the matrix form we can rearrange the terms
but we need to ensure that like indeces are adjacent 
to each other before contraction.  This is because
the product of 2 matrices is only defined if the
number of columns in the first matrix is equal to
the number of rows in the second matrix, i.e.

         (m x n)(n x m) = m x k

Therefore,  A^m_nBⁿ_m will correspond to a legal matrix 
operation while Bⁿ_mA^m_n will not.

Example:

M^μν = A^μ_σA^ν_ρM^σρ:

    = A^ν_ρA^μ_σM^σρ:

    = A^ν_ρB^μρ where B^μρ = A^μ_σM^σρ

    = A^ν_ρ(B^ρμ)^T 

    = (AB^T)^νμ

    = ((AB^T)^νμ)^T = (BA^T)^μν

But B = AM.  Therefore,

M' = AMA^T