Redshift Academy

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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

Electric Fields --------------- Electric field is defined as the electric force per unit charge. Thus: E = F/q or F = qE Point charge: E = kq/r2 The direction of the field is taken to be the direction of the force it would exert on a positive test charge. The electric field is radially outward from a positive charge and radially in toward a negative point charge. This is illustrated in the following examples. Colinear charges: A | B ------+-------o-------+------------+--- -0.02m ^ 0.02m 0.1m -5.5μC 2.5μC EA = k(5.5μC)/(.02)2 - k(2.5μC)/(.12 m)2 The negative charge term is positive because the electric field at A is pointing in the positive x direction. While the positive charges term is negative because the electric field is pointing in the negative x direction. EB = -k(5.5μC)/(.02)2 - k(2.5)/(.08)2 The negative charge term is negative because the electric field at B is pointing in the negative x direction. While the positive charges term is negative because the electric field is pointing in the negative x direction. Charges at corner of a square: A B -2q <-- d --> -3q . α ^ . | . d . | . v . +q -4q D C Force on A: x: FAB + FACcosα y: FAD + FACsinα B is pushing A to the left. D is pulling A downwards. Cx is pushing A upwards Cy is pushing A to the left. Infinite line of charge: Ez /| / | λ = charge per unit length r / | z / α | ---------------------------- x-> -a b dEz = (kλdx/r2)cosα = (kλdx/r2)(z/r) b = kλz∫dx/(z2 + x2)3/2 a = (kλ/z)[b/(z2 + b2) + a/(z2 + b2)] As a and b -> ∞ E = 2kλ/z = λ/2πzε0 Charged ring: Ez /| / | r/ |z / | / θ | ------------ Q R Ez = kQcosθ/r2 = kQz/r3 = kQz/(z2 + R2)3/2 The max field is when dE/dz = 0 => z = R/√2 Charged disc: R Ez = 2πkzσ∫R'dR'/(z2 + R'2)3/2 0 = 2πkσ[1 - z/(z2 + R2)] Gauss's Law ----------- The total electric flux ,φ, out of a closed surface is equal to the charge enclosed by the surface divided by the permittivity. In integral form: ∫E.dA = φ = q/ε0 φ is perpendicular to A. If not then φ = EAcosθ (dot product) Gauss's law is an alternative to Coulomb's law for calculating the electric field due to a given distribution of charge. Point charge: q ---->x Ex = kq/r2 Coulomb r φ = EA = q/ε0 E = q/ε0A = q/4πr2ε0 = kq/r2 Infinite line of charge: φ = EA = q/ε0 = λl/ε0 λ = charge per unit length E = λl/2πε0rl = λ/2πε0r Infinite sheet of charge (2 surfaces): ^ E | ------------ A φ = E2A = σA/ε0 | v E E = σ/2ε Electric Field Inside a Conductor --------------------------------- In steady state the electric field inside a conductor must be 0 otherwise charges would move around until they find an arrangement that makes the electric field equal to 0 in the interior. Any net charge must therefore reside on the surface. ^ E | ---------- A | E = 0 | ---------- The outward flux from the surface is EA The charge enclosed = σA/ε0 where σ is the charge per unit area Therefore, σA/ε0 = EA so E = σ/ε0 Ex 1. Suppose that a steam of negatively charged particles is blown through a circular pipe with radius r = 5.0cm. Assume that the charge is spread throughout the volume of particles in the pipe so that the charge density ρ is constant. (1) Find the formula for the electric field at the pipe surface. (2) Find the formula for the field at a distance r1 from the pipe surface. (1) EA = q/ε0 q = πr2lρ A = 2πrl => E = ρr/2ε0 (2) EA = q/ε0 q = πr2lρ A = 2πr1l => E = πr2lρ/2πr1ε0l = r2ρ/2r1ε0 Now, ρ = q/πr2l E = r2(q/πr2l)/2r1ε0 = (q/πl)/2r1ε0 = (q/l)/2πr1ε0 = λ/2πr1ε0 where λ is the charge per unit length. This equation is the same as the one for the field due to an infinite line of charge. Electric Potential Energy and Electric Potential (Voltage) ---------------------------------------------------------- Electric Potential energy, U, can be defined as the capacity for doing work which arises from position or configuration. U = force * distance = kQq/r = qE.r = qErcosθ In terms of calculus: ΔU = ∫F.dr = ∫Frcosθ x A q--- B \θ | h \ | y \ | \| C UAB = qEx UBC = 0 UAC = qEhcosθ = x Therefore, the trajectory doesn't matter - only displacment in the direction of the field is counted. If a positive charge moves in the same direction as the field then U will fall. (the field does work on the charge). If it moves against the field, U will rise (an external agent must do work on the charge). The opposite is true for a negative charge in the same field. The voltage difference (aka electrical potential), V, is defined as the work done per unit charge against an electric field. V = U/q = kQq/rq = kQ/r = (kQ/r2)r = Er Therefore, E = V/r and U = qV In terms of calculus: ΔV = dW/q = (1/q)∫F.ds = ∫E.ds = ∫Ecosθds E = -dV/dx Ex 1. Electron in electric field. How fast is it moving after distance s? F = qE = ma => a = qE/m v2 = 2as assume initial velocity = 0 Alternatively, if we were given the potential difference, V qV/s = ma => a = qV/ms v2 = 2as = (2qV/ms)s = 2qV/m Note: The final speed of the electron is entirely determined by its charge and the potential difference through which it is accelerated. We could also use the conservation of energy to solve the problem. U = qV = (1/2)mv2 v2 = 2qV/m Ex 2. Dipole. Find points where electric potential is a certain +V -Q +Q -+----------------------|-------------+---------+--------+------->x -d O A +d B The electric potential will look something like: || / \ -d / \ --------0-------------- \ / +d \ / || There will be 2 points, A and B, that satisfy this condition. VA = KQ/(x + d) - KQ/(d - x) = KQ{1/(x + d) - 1/(d - x)} = KQ{((x + d) - (d - x)/(x + d)(d - x)} VA/KQ = 2x/(d2 - x2) solve for x. VB = KQ/(x + d) - KQ/(x - d) = KQ{1/(x + d) - 1/(x - d)} = KQ{((x + d) - (x - d)/(x + d)(x - d)} VB/KQ = 2d/(x2 - d2) solve for x.