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Astronomy

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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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Stochiometry
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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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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions .
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Electric Fields
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Error Analysis
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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity .
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases .
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Kirchoff's Laws
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Maxwell's Equations .
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation .
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Pascal's Principle
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Phase and Group Velocity
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Poiseuille's Law
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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Young's Modulus

Climate Change

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Keeling Curve .

Cosmology

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Baryogenesis
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Cosmic Background Radiation and Decoupling .
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CPT Symmetries
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Dark Matter .
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Friedmann-Robertson-Walker Equations .
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes .
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Olbers' Paradox .
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Planck Units .
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Stephen Hawking's Last Paper .
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Stephen Hawking's PhD Thesis .
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The Big Bang Model
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Vacuum Energy .

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets .
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

Game Theory

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The Truel .

General Relativity

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Basis One-forms .
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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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Geodesics
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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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Ricci Flow .
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Stress-Energy-Momentum Tensor .
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Tensors
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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 Induced Metric
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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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Euler-Lagrange Equation .
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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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Symmetry and Conservation Laws - Noether's Theorem .

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series .
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Asymptotes
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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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Fractals
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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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Inequalities
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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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Related Rates
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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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Symbolic Logic
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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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Unit Vectors
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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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Data Types
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Expected Value
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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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One-Way ANOVA
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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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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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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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The Qubit .

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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Bohr Atom
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Clebsch-Gordan Coefficients .
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Commutators
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Dyson Series
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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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Wavepackets

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 MOSFET
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The P-N Junction

Special Relativity

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4-vectors .
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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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The Ideal Gas

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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BCS Theory
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Cooper Pairs
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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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Constants
Last modified: February 9, 2022 ✓

Electromagnetic (Faraday) Tensor -------------------------------- Electromagnetic 4 - Potential: Aμ = (A0,A1,A2,A3) = (φ/c,A1,A2,A3) Where φ is the electric potential and A is the magnetic vector potential. 4-gradient: ∂μ = (∂0,∂1,∂2,∂3) = ((1/c)∂/∂t,∂/∂x,∂/∂y,∂/∂z) Gauge Invariance ---------------- We can define an asymmetric tensor, Fμν, which is GAUGE INVARIANT as follows: Fμν = ∂μAν - ∂νAμ Proof: Gauge field: Aμ -> Aμ - (1/e)∂μθ(x) Dμφ = ∂μφ + ieAμφ ∂μAν -> ∂μAν - (1/e)∂μνθ ∂νAμ -> ∂νAμ - (1/e)∂νμθ) Fμν = ∂μAν - ∂νAμ => itself Q.E.D Consider F01: F01 = ∂0A1 - ∂1A0 = (1/c)∂0A1 - ∂1A0 = (1/c)∂0A1 - ∂1(-φ/c) = (1/c){∂Ax/∂t + ∂φ/∂x} Now, the electric field is defined as: E = -∂A/∂t - ∇φ = (-∂Ax/∂t - ∂φ/∂x) + (-∂Ay/∂t - ∂φ/∂y) + (-∂Az/∂t - ∂φ/∂z) Comparing this to F01 reveals: F01 ≡ -Ex/c Now consider F12: F12 = ∂1A2 - ∂2A1 Now, the magnetic field is defined as: B = ∇ x A = (∂2A3 - ∂3A2) + (∂3A1 - ∂1A3) + (∂1A2 - ∂2A1) Comparing this to F12 reveals: F12 ≡ Bz The components of Fμν are: F00 = ∂0A0 - ∂0A0 = 0 F01 = ∂0A1 - ∂1A0 = -Ex/c F02 = ∂0A2 - ∂2A0 = -Ey/c F03 = ∂0A3 - ∂3A0 = -Ez/c F10 = ∂1A0 - ∂0A1 = Ex/c F11 = ∂1A1 - ∂1A1 = 0 F12 = ∂1A2 - ∂2A1 = Bz F13 = ∂1A3 - ∂3A1 = -By F20 = ∂2A0 - ∂0A2 = Ey/c F21 = ∂2A1 - ∂1A2 = -Bz F22 = ∂2A2 - ∂2A2 = 0 F23 = ∂2A3 - ∂3A2 = Bx F30 = ∂3A0 - ∂0A3 = Ez/c F31 = ∂3A1 - ∂1A3 = By/c F32 = ∂3A2 - ∂2A3 = -Bx F33 = ∂3A3 - ∂3A3 = 0 Therefore, we can write (note by convention the B components are written as the -ve of the actual values). - - | 0 Ex/c Ey/c Ez/c | |-Ex/c 0 -Bz By | = Fμν |-Ey/c Bz 0 -Bx | |-Ez/c -By Bx 0 | - - This is COVARIANT ELECTROMAGNETIC TENSOR. - - | 0 -Ex/c -Ey/c -Ez/c | | Ex/c 0 -Bz By | = Fμν | Ey/c Bz 0 -Bx | | Ez/c -By Bx 0 | - - This is CONTRAVARIANT ELECTROMAGNETIC TENSOR. Note: Fμν = ημαηνβFαβ and Fμν = ημαηνβFαβ Properties: Fμν = -Fνμ ... antisymmetric Fμν is gauge invariant but not Lorentz invariant. However, the inner product FμνFμν is a scalar quantity which is Lorentz invariant, i.e. FμνFμν = 2(B2 - E2/c2) Maxwell's Equation in Flat Spacetime ------------------------------------ ∇.E = ρ/ε and ∇ x B = μ0J + (1/c2)∂E/∂t Reduce to: ∂Fμν/∂xμ = ∂μFμν = μ0jν Where jν = ρ∂xν/∂t = (cρ, j) ... the 4-current ∇.B = 0 and ∇ x E = -∂B/∂t reduce to: ∂Fμν/∂xγ + ∂Fνγ/∂xμ + ∂Fγμ/∂xμ = ∂γFμν + ∂μFνγ + ∂νFγμ = 0 Maxwell's Equation in Curved Spacetime -------------------------------------- Replace the partial derivative with the covariant derivative to get: ∇μFμν = μ0JνγFμν + ∇μFνγ + ∇νFγμ = 0 This is equal to: (∂γFμν - ΓσγμFσν - ΓσγνFμσ) + (∂μFνγ - ΓσμνFσγ - ΓσμγFνσ) + (∂νFγμ - ΓσνγFσμ - ΓσνμFγσ) = 0 We can use the antisymmetric properties of Fab = -Fba to get: (∂γFμν - ΓσγμFσν - ΓσγνFμσ) + (∂μFνγ - ΓσμνFσγ + ΓσμγFσν) + (∂νFγμ + ΓσνγFμσ ΓσνμFσγ) = 0 We can use the symmetry of the Christofel symbols, Γσαβ = Γσβα to get: ∂γFμν + ∂μFνγ + ∂νFγμ - ΓσγμFσν + ΓσγμFσν - ΓσγνFμσ + ΓσγνFμσ - ΓσμνFσγ + ΓσμνFσγ = 0 Which simplifies to: ∂γFμν + ∂μFνγ + ∂νFγμ = 0