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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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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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Gauss's Law of Universal Gravity
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Gravity - Force and Acceleration
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Kinematics
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Maxwell's Equations
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Moments and Torque
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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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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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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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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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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: January 16, 2020

One Forms --------- Basis One-forms --------------- A one-form is like a machine that when fed with a vector spits out a number which depends linearly on the input. We can define dxμ to be the basis one-form which gives 1 when you input the basis tangent vector, ∂μ, and 0 when you input ∂ν for μ ≠ ν. Thus, [dxμ](∂ν) = (∂/∂xν)(xμ) = ∂xμ/∂xν = δμν This should not be interpreted as ∂[∂ν]/∂xμ but rather as (dxμ) 'acting' on (∂ν). The basis one-forms are the gradients of the scalar coordinate functions (i.e. xμ = xμ(p) where (U,p) is a local chart. Equivalently we can say φ: U -> Rn where φ consists of n real valued functions, xμ). We can show this as follows: ∇ := (∂/∂xν)eν ∇xμ = (∂xμ/∂xν)eν Where ∂xμ/∂xν are the components of the gradient (a dual vector) and eν are the dual basis vectors. Therefore, ∇xμ = δμνeν = eμdxμ Transformation Laws ------------------- vectors one-forms ------- --------- ∂μ -> ∂'μ = (∂xμ/∂x'μ)∂μ dxμ -> dx'μ = (∂x'μ/∂xμ)dxμ Vμ -> V'μ = (∂x'μ/∂xμ)Vμ ωμ -> ω'μ = (∂xμ/∂x'μμ V'x = (∂x'/∂x)Vx + (∂x'/∂y)Vy V'y = (∂y'/∂x)Vx + (∂y'/∂y)Vy - - - - - - | V'x | = | ∂x'/∂x ∂x'/∂y || Vx | | V'y | | ∂y'/∂x ∂y'/∂y || Vy | - - - - - - Vector components have a contravariant transformation law. ∂x' = (∂x/∂x')∂x + (∂y/∂x')∂yy' = (∂x/∂y')∂x + (∂y/∂y')∂y - - - - - - | ∂x'y' | = | ∂xy || ∂x/∂x' ∂x/∂y' | -    -    -   - | ∂y/∂x' ∂y/∂y' | - - Basis vectors have a covariant transformation law. ∂φ/∂x' = (∂x/∂x')(∂φ/∂x) + (∂y/∂x')(∂φ/∂y) ∂φ/∂y' = (∂x/∂y')(∂φ/∂x) + (∂y/∂y')(∂φ/∂y) - - - - - - | ∂φ/∂x' ∂φ/∂y' | = | ∂φ/∂x ∂φ/∂y || ∂x/∂x' ∂x/∂y' | - - - - | ∂y/∂x' ∂y/∂y' | - - One-form components have a covariant transformation law. dx' = (∂x'/∂x)dx + (∂x'/∂y)dy dy' = (∂y'/∂x)dx + (∂y'/∂y)dy - - - - - - | dx' | = | ∂x'/∂x ∂x'/∂y || dx | | dy' | | ∂y'/∂x ∂y'/∂y || dy | - - - - - - Basis one-forms have a contravariant transformation law. One-forms --------- One-forms are also known as covariant/dual vectors. They are often written as ω: df ≡ ω = ωμdxμ Where, ωμ = {∂f/∂x1,∂f/∂x2, ...} and, dxμ = {dx1,dx2, ...} Therefore, df = ω = (∂f/∂x1)dx1 + (∂f/∂x2)dx2 ... = (∂f/∂xμ)dxμ Example: Conversion one-form in polar coorinates to one-form in xy plane. x = rcosθ y = rsinθ dx = (∂x/∂r)dr + (∂x/∂θ)dθ dy = (∂y/∂r)dr + (∂y/∂θ)dθ dx = cosθdr - rsinθdθ dy = sinθdr + rcosθdθ Likewise, dr = (∂r/∂x)dx + (∂r/∂y)dy dθ = (∂θ/∂x)dx + (∂θ/∂y)dy ∂r/∂x = x/√(x2 + y2) = x/r = cosθ ∂r/∂y = y/√(x2 + y2) = y/r = sinθ ∂θ/∂x = -y/(x2 + y2) = -y/r2 = -sinθ/r ∂θ/∂y = x/(x2 + y2) = x/r2 = cosθ/r dr = (x/r)dx + (y/r)dy dθ = (-y/r2)dx + (x/r2)dy The product of a one-form with a vector is given by: v = v1(∂/∂x'1) + v2(∂/∂x'2) + ... = v11 + v21 + ... = Σvμμ μ ω = ω1dx1 + ω2dx2 + ... = Σωμdxμ ν vω = Σvμμ(Σωνdxν) μ ν = Σvμωνδνμ μν = Σvμωμ ∈ R μ Example: Product of vector and one-form in polar coordinates. dx = cosθdr - rsinθdθ dy = sinθdr + rcosθdθ ∂u/∂x = (∂u/∂r)(∂r/∂x) + (∂u/∂θ)(∂θ/∂x) = cosθ(∂u/∂r) - (1/r)sinθ(∂u/∂θ) ∂u/∂y = (∂u/∂r)(∂r/∂y) + (∂u/∂θ)(∂θ/∂y) = sinθ(∂u/∂r) + (1/r)cosθ(∂u/∂θ) (cosθdr - rsinθdθ)(cosθ(∂u/∂r) - (1/r)sinθ(∂u/∂θ)) = cos2θdr(∂u/∂r) - (1/r)cosθsinθdr(∂u/∂θ) - rsinθcosθdθ(∂u/∂r) + sin2θdθ(∂u/∂θ) = 1 (sinθdr + rcosθdθ)(sinθ(∂u/∂r) + (1/r)cosθ(∂u/∂θ)) = sin2θdr(∂u/∂r) + (1/r)sinθcosθdr(∂u/∂θ) - rcosθsinθdθ(∂u/∂r) + cos2θdθ(∂u/∂θ) = 1 Exterior Derivative ------------------- df(V) := Vf = Vμμf = (∂f/∂xμ)Vμ The reasoning for this can be found by considering the directional derivative d/dλ = (dxμ/dλ)∂μ. df[(dxμ/dλ)(∂μ)] = df[(dxμ/dλ)(∂/∂xμ)] ---------- ^ = df[(d/dλ)] | V = (d/dλ)[f] = df/dλ In other words, df when given a vector returns the directional derivative in that direction. It is worth noting that f is a scalar function (0-form). df is therefore a 1-form (dual vector). Therefore, df acting on the tangent vector dxμ/dλ)(∂μ) produces a scalar, df/dλ. Example: f(x,y) = x2y at point (1,1) in the direction (-1,1). = 2xy + x2 = (2,1) at (1,1) (-1,1) => (-√2/2,√2/2) after normalization (2,1).(-√2/2,√2/2) = -√2/2 ^ ^ ^ | | | ∂f/∂xμ dxμ/dλ df/dλ = ∂μf = Vμ In terms of the one-form we get: df(V) = Vμ(∂f/∂xμ) = (∂f/∂xμ)Vμ = 2xy(-√2/2) + x2(√2/2) = -√2/2 at (1,1) as before. Infinitesimal Version of the Chain Rule --------------------------------------- From before we had: df(V) = (∂f/∂xμ)Vμ Now, dxμ(V) = dxμ(Vνν) = Vνdxμ(∂ν) = Vνδμν = Vμ Therefore, df(V) = (∂f/∂xμ)dxμ(V) Or, df = (∂f/∂xμ)dxμ In the context of one-forms, dx = (∂x/∂xμ)dxμ is the infinitesimal version of the chain rule. We can see the relationship as follows: df/dxμ = (∂f/∂g(xμ))(dg(xμ)/dxμ) where f = f(g(xμ)) Therefore, rearranging: df/dg(xμ) = (∂f/∂xμ)(dxμ/dg(xμ)) multiply by dg(xμ) to get: df = (∂f/∂xμ)dxμ Strictly speaking when we talk about infinitesimal displacements we should be specifying differential forms in the equations and not dxμ. This is because displacement is a scalar that can only be obtained by contracting a one-form with a tangent vector. For example, the line segment, ds2 = ημνdxμdxν, should really be ds2 = ημνdxμdxν. However, most references do not distinguish between 'dx', the informal notion of an infinitesimal displacement, and 'dxμ', the rigorous notion of a basis one-form given by the gradient of a coordinate function. Example: (ds)2 = (dx)2 + (dy)2 = ((∂x/∂r)dr + (∂x/∂θ)dθ)2 + ((∂y/∂r)dr + (∂y/∂θ)dθ)2 = (cosθdr - rsinθdθ)2 + (sinθdr + rcosθdθ)2 = cos2θ(dr)2 + r2sin2θ(dθ)2 + sin2θ(dr)2 + r2cos2θ(dθ)2 = (dr)2 + r2(dθ)2