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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 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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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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Penrose Diagrams
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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

Game Theory

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The Truel .

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 Computing

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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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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 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 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

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: January 26, 2018

Contravariant and Covariant Components of a Vector -------------------------------------------------- Consider cartesian coordinates. ^ X2 | | - x2|......c - | / |c| = L | /. | L/ . | / . | / . |/ . --------------> X1 x1 L2 = (x1)2 + (x2)2 Now how do we get L when we have non-cartesian coordinates? We need to introduce the concept of contravariant and covariant components. In cartesian coordinates the covariant and contravariant components are the same. Contravariant components: u1 = x1 - x2/tanα u2 = x2/sinα Covariant components: u1 = x1 u2 = Lcos(α - β) = Lcos(α)cos(β) + Lsin(α)sin(β) Now Lcosβ = x1 and Lsinβ = x2 u2 = x1cosα + x2sinα To get L2 try: u1u1 = x1(x1 - x2/tanα) = (x1)2 - (x1x2/tanα) u2u2 = (x1cosα + x2sinα)x2/sinα = (x1x2/tanα) + (x2)2 Therefore, u1u1 + u2u2 = (x1)2 + (x1)2 Tranformation Matrices ---------------------- We can write: - - - - | u1 | = H | x1 | | u2 | | x2 | - - - - where - - - - H = | ∂u1/∂x1 ∂u1/∂x2 | = | 1 -1/tanα | | ∂u2/∂x1 ∂u2/∂x2 | | 0 1/sinα | - - - - and - - - - | u1 | = M | x1 | | u2 | | x2 | - - - - where - - - - M = | ∂u1/∂x1 ∂u1/∂x2 | = | 1 0 | | ∂u2/∂x1 ∂u2/∂x2 | | cosα sinα | - - - - H and M are transformation matrices (JACOBEAN MATRICES) How do we convert between the covariant and contravariant components and vice versa? - - - - | x1 | = H-1 | u1 | | x2 |   | u2 | - - - - and - - - - | x1 | = M-1 | u1 | | x2 |   | u1 | - - - - where H-1 and M-1 are the inverse of the JACOBEAN MATRICES H and M. So,    - - - - H-1 | u1 | = M-1 | u1 |   | u2 |   | u2 |    - - - - Therefore, - - - - | u1 | = HM-1 | u1 | | u2 |   | u2 | - - - - So, - - - - | u1 | = gij | u1 | | u2 |   | u2 | - - - - where gij is the INVERSE METRIC tensor. In summary, we can construct a map as follows: u1 u2    / ^  ^    / /  | | M-1/ /M  | |    / /   | |    v /   | |   x1 gij| |gij   x2   | |    ^ \   | |    \ \   | | H-1\ \H  | |    \ \  | |    \ v   v u1 u2 From the map: - - gij = HM-1 where M-1 = (1/sinα)| sinα 0 |       | -cosα 1 | - - Therefore, - - gij = (1/sin2α)| 1 -cosα |    | -cosα 1 | - - Now gij = (gij)-1 (= H-1M from the map). Where gij is the METRIC TENSOR. Therefore, - - gij = (1/det[gij])| 1/sin2α -cosα/sin2α |      | -cosα/sin2α 1/sin2α | - - det[gij] = 1/sin4α - cos2α/sin4α = sin2α/sin4α = 1/sin2α So, - - gij = sin2α| 1/sin2α cosα/sin2α |    | cosα/sin2α 1/sin2α | - - - - = | 1 cosα | | cosα 1 | - - - - - - Now gijgij = (1/sin2α)| 1 -cosα || 1 cosα |      | -cosα 1 || cosα 1 | - - - - - - = (1/sin2α)| 1 - cos2α cosα - cosα |  |-cosα + cosα -cos2α + 1 | - - - - = | 1 0 | | 0 1 | - - Now gij = H-1M from the map so M = gijH - - - - = | 1 cosα || 1 -1/tanα | | cosα 1 || 0 1/sinα | - - - - - - = | 1 (-1/tanα + cosα/sinα) | | cosα (-cosα/tanα + 1/sinα) | - - - - = | 1 0 | which agrees with before. | cosα sinα | - - Similarly, gij = HM-1 from the map so H = gijM H = gijM - - - - = 1/sin2α | 1 -cosα || 1 0 |  |-cosα 1 ||cosα sinα | - - - - - - = | 1 -1/tanα | which agrees with before. | 0 1/sinα | - - In summary, the metric tensor enables us to convert the contravariant components of a vector to their covariant form, and vice versa, in the following manner: - - - - 2 gij | u1 | = | u1 | <--> Σgijuj = ui   | u2 | | u2 | j=1 - - - - and - - - - 2 gij | u1 | = | u1 | <--> Σgijuj = ui   | u2 | | u2 | j=1 - - - - This process of going between contravariant and covariant components, and vice versa, is referred as the "raising and lowering of indeces".