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Astronomy

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Celestial Coordinates
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Celestial Navigation
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Distance Units
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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 Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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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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Laplace's and Poisson's Equations
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Lorentz Force Law
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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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Planck Radiation Law
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Poiseuille's Law
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Radioactive Decay
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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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Stefan-Boltzmann Law
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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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Wien's Displacement Law
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Young's Modulus

Climate Change

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Keeling Curve

Cosmology

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Penrose Diagrams
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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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Geometries of the Universe
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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

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

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Gravitational Time Dilation
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Gravitational Waves
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One-forms
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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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Ricci Decomposition
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Ricci Flow
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Stress-Energy Tensor
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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 Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Metric Tensor
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Vierbein (Frame) Fields
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World Lines Refresher

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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Basic Group Theory
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Basic Representation Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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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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Sets, Groups, Modules, Rings and Vector Spaces
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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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Symmetric Groups
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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 Integers Modulo n Under + and x
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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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Vector Calculus
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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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
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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots
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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 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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Basic Relationships
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Bell's Theorem
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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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Entangled States
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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 Exclusion Principle
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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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Schrodinger Wave Equation (continued)
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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 Observer Effect
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The Qubit
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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 4 - Potential
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Energy and Momentum, E = mc2
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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 Lorentz Group
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Velocity Addition

Statistical Mechanics

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Black Body Radiation
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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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BCS Theory
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Introduction to Superconductors
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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

test

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test

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
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Formulas
Last modified: March 30, 2020

Classical Field Theory ---------------------- Massless Wave Equation ---------------------- A field in physics may be envisioned as if space were filled with interconnected vibrating masses and springs, and the strength of the field is like the displacement of a ball from its rest position. The theory requires "vibrations" in, or more accurately changes in the strength of, such a field to propagate as per the appropriate wave equation for the particular field in question. Consider an array of masses, m, separated by distance ε and connected by massless springs each with spring contant, k. φ ^ | | o ki | ^ o o/\/\/o | | | o | o v o |o | o o | v |________________________________________ x <-ε-> Let ε be a small interval. Let mi = εm and ki = K/ε. Thus, as ε -> 0 mi -> 0 and ki -> ∞. Consider oscillations in the vertical direction. . KE, T = Σi(1/2)εmφi2 PE, U = (K/2ε)Σii+1 - φi)2 Here it is assumed that the only contribution is from neighbouring particles. Note that spring force is inversely proportional to distance. Lagrangian, L = T - V . = Σi(1/2)εmφi2 - (K/2ε)Σii+1 - φi)2 . . ∂L/∂φi = εmφ The PE term requires a little more calculation: U = (K/2ε)[(φi+1 - φi)2 + (φi-1 - φi)2] -∂U/∂φi = -(K/ε)[(φi - φi+1) + (φi - φi-1)] = (K/ε)[-φi + φi+1 - φi + φi-1] = (K/ε)[(φi+1 - φi) - (φi - φi-1)] Apply the E-L equations to get the equations of motion: . d/dt(∂L/∂φi) - ∂L/∂φi = 0 .. εmφi = (K/ε)[(φi+1 - φi) - (φi - φi-1)] .. φi = (K/mε2)[(φi+1 - φi) - (φi - φi-1)] In the limit ε -> 0, φi -> φ(x) and the RHS is nothing more than the double derivative. Thus, .. φ = (K/m)∂2φ/∂x2 Write v2 = K/m where v is the velocity along the spring then, ∂2φ/∂t2 - v22φ/∂x2 = 0 This is the MASSLESS WAVE EQUATION. Note the similarity between the wave equation form and the invariant interval. Lagrangians for Classical Fields -------------------------------- A physical field can be thought of as the assignment of a physical quantity at each point of space and time. A field theory tends to be expressed mathematically by using Lagrangians. The Lagrangian for classical systems is a function of generalized coordinates q and dq/dt. In this view, time is an independent variable and q and dq/dt are dependent variables in phase space. The action for such a system is written as: A = ∫dt L(q,dq/dt) This formalism was generalized to handle field theory. In field theory, the dependent variables q and dq/dt are replaced by the value of a field at that point in spacetime. Thus, q -> φ and dq/dt -> dφ/dt. The action in this case is written as: Aφ = ∫dtd3x ℒ(φ,∂μφ) where ∂μ = ∂/∂xμ. Where ℒ is the LAGRANGIAN DENSITY. Consequently, L = ∫dx3 ℒ Euler-Lagrange Equations for a Field ------------------------------------ Consider: φ(x,t) = φ(x,t) + δφ(x,t) Where, δφ(x,t) = εf(x,t) and δφμ = ε∂f(x,t)/∂xμ Note: We have introduced the notation φμ ≡ ∂μφ δA = ε∫d4x[(∂ℒ/∂φ)f(x,t) + (∂ℒ/∂φμ)(∂f(x,t)/∂xμ)] = 0 Integrate by parts to get: δA = ε∫dtd3x[∂ℒ/∂φ - ∂/∂xμ(∂ℒ/∂φμ)]f(x,t) = 0 Therefore, for this to be true for a non-zero f(x,t), we must have: ∂/∂xμ(∂ℒ/∂φμ) - ∂ℒ/∂φ = 0 This is the EULER-LAGRANGE equation for a field. In 1 dimension this is: ∂/∂t(∂ℒ/(∂φ/∂t) - ∂/∂x(∂ℒ/(∂φ/∂x) ) - ∂ℒ/∂φ = 0 In general this is written as: ∂μ(∂ℒ/∂(∂μφ)) - ∂ℒ/∂φ = 0 The momenta canonically conjugate to the coordinates of the field (or simply the canonical momentum) is defined as: . π = ∂ℒ/∂φ Recall that the Lagrangian for the massless wave equation is: . L = Σi(1/2)εmφi2 - (K/2ε)Σii+1 - φi)2 . = ∫(1/2)φ2dx - (K/2)Σiε(φi+1 - φi)(φi+1 - φi)/ε2 . = ∫(1/2)φ2dx - (K/2)Σiε(∂φ/∂x)2 . = ∫(1/2)φ2dx - (K/2)∫(∂φ/∂x)2]dx . = ∫[(1/2)φ2dx - (K/2)(∂φ/∂x)2]dx = ∫ℒdx Where, . ℒ = (1/2)φ2 - (K/2)(∂φ/∂x)2 ℒ is a scalar and is, therefore, Lorentz invariant. Now ℒ may also contain additional terms to the potential energy. For example, the system may be in a gravitational field. This term can be defined as follows: V(φ) = (μ2/2)φ2 Therefore, ℒ becomes: . ℒ = (1/2)[φ2 - v2(∂φ/∂x)2 - μ2φ2] Or, in shorthand form: ℒ = (1/2)[-∂μφ∂μφ - μ2φ2] Where, ∂μ = (∂t,∂x,∂y,∂z) and, ∂μ = (-∂t,∂x,∂y,∂z) Apply E-L equations to get: ∂{∂((1/2)(∂φ/∂t)2}/∂t = ∂2φ/∂t2 ∂{∂((1/2)(∂φ/∂x)2}/∂x = ∂2φ/∂x2 ∂ℒ/∂φ = μ2φ Therefore, ∂2φ/∂t2 - v22φ/∂x2 + μ2φ = 0 \ \ Field KE Field PE Without the μ term this becomes: ∂2φ/∂t2 - v22φ/∂x2 = 0 So, not surprisingly, solutions to the E-L equation for a simple field is the massless wave equation. Solutions of this equation are of the form: φ = f(x + vt) + g(x - vt) waves with right and left moving parts. Note that wave shapes are preserved as they pass through each other. If the m term is added then the waves scatter each other. To appreciate the content of this equation, consider the connection with quantum mechanics: p = -ih∂/∂x ∴ ∂2φ/∂x2 = p2/h2 E = (ih)∂/∂t ∴ ∂2φ/∂t2 = -E2/h2 We can write: (E2 - c2p2 - m2c4)φ = 0 with m2c4/h2 = μ2 This is the Klein-Gordon equation.