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 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: June 13, 2022 ✓

Ricci Decomposition ------------------- The Ricci decomposition produces an irreducible representation of the Riemann tensor that consists of 3 different parts. Rμναβ = Sμναβ ⊕ Eμναβ ⊕ Cμναβ Where: Sμναβ = (R/n(n - 1))(gμαgβν - gμβgαν) where R = gμνRμν and Rμν is the contraction of the 1st and 3rd indeces of Rβμαν, i.e. Rμν = Rαμαν = δραRαμρν = δραgλαRλμρν = gρλRλμρν = Rμν. Eμναβ = (1/(n - 2))(gμαZνβ - gμβZνα + gνβZμα - gναZμβ) where Zμν = Rμν - (1/n)gμνR is the traceless Ricci tensor. Eμναβ is semi-traceless. Semi-traceless means it is built from the metric (which is not traceless) and the traceless Ricci tensor which is. Cμναβ is a the WEYL TENSOR. Therefore, the first piece, the scalar part, is built out of the curvature scalar R and the metric. The second piece, the semi-traceless piece, is built out of the metric and the traceless Ricci tensor. The third piece is what is left over and is called the Weyl tensor. This final piece is completely traceless. The Riemann Tensor ------------------ Properties: Rμναβ = Rαβμν ... symmetric      = -Rνμαβ ... antisymmetric      = -Rμνβα ... antisymmetric Rμναβ + Rμβbα + Rμαβν = 0 ... 1st BIANCHI IDENTITY. ∇ρRμναβ + ∇αRμνβρ + ∇βRμνρα = 0 ... 2nd BIANCHI IDENTITY. The Ricci Tensor ---------------- The Ricci tensor represents that part of the gravitational* field which is due to the immediate presence of nongravitational energy and momentum. It describes how the volume of a body changes due to tidal forces it feels when moving along a geodesic. Converging geodesics lead to a shrinking of the volume. Diverging geodesics lead to an expanding volume. * There is no term for gravitational energy in the Stress-Energy tensor. The stress-energy tensor is defined locally and only includes non-gravitational forms of energy (matter) whereas gravitational energy is a non-local quantity. This argument comes from the Equivalence Principle. One can always find in any given locality, a frame of reference in which all local gravitational fields vanish (i.e. a free falling reference frame). In other words, in an appropriate coordinate system a small region of space is free of a gravitational field. GR states that gravitational energy exists globally but is not localizable. There is no possibility of defining an energy density of the gravitational field in GR at one point. Properties: - Using the property that Rβμνα = Rναβμ Rνμ = gβαRανβμ ... the metric is symmetric. Therefore, Rμν = Rνμ ... symmetric The Scalar Curvature -------------------- The Scalar curvature assigns a single real number determined by the intrinsic geometry of spacetime near that point. Specifically, it represents the amount by which the volume of a geodesic ball (a ball made of a series of flat sides) in a curved spacetime, VS, deviates from that of the standard ball in Euclidean space, VE. The ratio of the 2 volumes is given by: VS/VE = 1 - Rr2/6(n + 2) When the scalar curvature is positive at a point, the volume of the ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is larger than it would be in Euclidean space. To reiterate, the difference between the Ricci tensor and the Scalar curvature is subtle. The former shows how an arbitrary volume changes along a geodesic. The latter measures the deviation of the volume of a geodesic ball from its volume in Euclidean space. The Weyl Tensor --------------- The Weyl tensor is associated with changes in the shape of a body due to the tidal forces it feels when moving along a geodesic. It does not effect the volume. For example, a ball moving along a geodesic could be stretched into an ellipsoid with the same volume as the ball. Properties: Cμναβ = Cαβμν ... symmetric      = -Cνμαβ ... antisymmetric      = -Cμνβα ... antisymmetric Cμναβ + Cμβbα + Cμαβν = 0 ... 1st BIANCHI IDENTITY. ∇ρCμναβ + ∇αCμνβρ + ∇βCμνρα = 0 ... 2nd BIANCHI IDENTITY. The Weyl tensor is completely traceless. The 3 Tensors ------------- ------------------------------------------ | Riemann | Ricci | Weyl | +-----+----+----+----+---+---+----+---+---+ Dimension (n) | 4 | 3 | 2 | 4 | 3 | 2 | 4 | 3 | 2 | +-----+----+----+----+---+---+----+---+---+ Total Comps. | 256 | 81 | 16 | 16 | 9 | 1 | | | | +-----+----+----+----+---+---+----+---+---+ Indep. Comps. | 20 | 6 | 1 | 10 | 6 | 1 | 10 | 0 | 0 | ----------------------------------------- Notes: - The Riemann curvature tensor is a sum of the Weyl curvature tensor plus terms built out of the Ricci tensor and Scalar curvature. - In 4D it is both necessary and sufficient for the Riemann tensor to vanish for the manifold to be flat. If the Ricci tensor vanishes the spacetime is said to be Ricci flat. Ricci flat spacetime will still have curvature unless the Weyl tensor vanishes. Regions of spacetime in which the Weyl tensor vanishes contain no gravitational radiation and are conformally flat. This means that each point has a neighborhood that can be mapped to flat space by an angle preserving transformation. - In 3D it is both necessary and sufficient for the Ricci tensor to vanish for the manifold to be flat. - In 2D it is both necessary and sufficient for the Scalar tensor to vanish for the manifold to be flat. - The Ricci tensor can describe everything about 3D curvature and all the 81 elements of the Riemann tensor can be calculated from the 6 independent components of the Ricci tensor. - Independent components are calculated from: Riemann = n2(n2 - 1)/12 Ricci = n(n + 1)/2 (because of symmetry) Weyl = n2(n2 - 1)/12 - n(n + 1)/2 for n > 2