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Astronomy

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Celestial Coordinates
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Celestial Navigation
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Distance Units
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Location of North and South Celestial Poles

Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions
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Electric Fields
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Error Analysis
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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases
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Laplace's and Poisson's Equations
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Lorentz Force Law
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Maxwell's Equations
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation
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Pascal's Principle
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Phase and Group Velocity
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Planck Radiation Law
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Poiseuille's Law
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Radioactive Decay
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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Stefan-Boltzmann Law
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect
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Wien's Displacement Law
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Young's Modulus

Climate Change

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Keeling Curve

Cosmology

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Penrose Diagrams
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Baryogenesis
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Cosmic Background Radiation and Decoupling
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CPT Symmetries
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Dark Matter
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Friedmann-Robertson-Walker Equations
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Geometries of the Universe
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes
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Olbers' Paradox
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Planck Units
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Stephen Hawking's Last Paper
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Stephen Hawking's PhD Thesis
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The Big Bang Model

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Gravitational Time Dilation
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Gravitational Waves
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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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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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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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Quaternions 1
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Quaternions 2
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Regular Polygons
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Sets, Groups, Modules, Rings and Vector Spaces
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Spherical Trigonometry
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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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Chebyshev's Theorem
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Pearson Correlation
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Permutations and Combinations
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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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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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 26, 2018

Definition of Entropy --------------------- Entropy is a measure of our ignorance regarding the exact state of a system. Consider the example of a gas in a bottle. The smaller the bottle the less uncertainty there is about where the molecules will be and so the entropy will be lower. On the other hand for a constant sized bottle the entropy will increase if heat is applied. This is because there will now be a wider range of possible velocities for the molecules in the gas. Now, if the bottle is opened in outer space, the gas will escape an diffuse into the vacuum. Now there are progressively more and more states that the gas can be in and so the entropy increases. Consequenty, as the system evolves in time the system becomes more and more disordered and our ignorance regarding the exact state of the system increases. Definition: We know state of system is somewhere in region defined by the distribution Pi Pi | | . . | . . | . . | . . | . . -------------------------------------------- State, i Then, S = -ΣiPilnPi This is the average value of lnPi Consider a specific case where all the states have the same Pi: Pi | | | ---------- | | | | | | --------------------------- State, i <-- M --> S = -ΣiPilnPi S = -Σi(1/M)ln(1/M) S = -M(1/M)ln(1/M) S = -ln(1/M) S = -lnM Thus, in this case, S is just the number of states in the system. Consider N coins: N = 2n S = lnN = nln(2) Entropy -------- The following discussion refers to distinguishable particles. Two particles can be considered to be distinguishable if their separation is large compared to their DeBroglie wavelength. For example, the condition of distinguishability is met by molecules in an ideal gas under ordinary conditions. On the other hand, two electrons in the first shell of an atom are inherently indistinguishable because of the large overlap of their wavefunctions. Consider N particles. Let ni = number of particles in the ith energy state, Ei Energy | + + +<- ith state with Energy Ei + + + + | The number of possible ways to fit the particles in the number of avalable states is called the multiplicity function. The multiplicity function for the whole system, W, is the product of the multiplicity functions for each energy Ei: W = Πi(N!/ni) ... 1. Consider the following example of how 6 particles can be distributed amongst 10 energy states. For the 3 states shown above the numbers of ways each state can be filled with 6 particles is given by: 6!/5!1! = 6 6!/3!2!1! = 60 6!/2!2!1!1! = 180 We can simplify 1. above by taking the log of both sides, lnW = lnN! - Σiln(ni!) Now apply Stirling's approximation i.e. lngA! = AlnA - A So lnW = NlnN - N - Σiniln(ni) + Σini Now, Pi = ni/N = probability that a given particle is in state i Therefore, lnW = NlnN - N - ΣiNPilnNPi + N = NlnN - (ΣiNPilnN + ΣiNPilnPi) = NlnN - (NlnN + NΣiPilnPi) = -NΣiPilnPi = NS where S = ΣiPilnPi This is the definition of ENTROPY. Boltzmann Distribution ------------------------ From the discussion of entropy it was shown that: lnW = NS = NΣiPilnPi where W is the mutiplicity function. Now maximize lnW by maximizing S. This can be achieved by using the method of Lagrange multipliers under the constraints ΣPi = 1 and E = ΣiPiEi (the average energy). Thus, lnW = -NΣiPilnPi - αΣiPi - βΣiEiPi Now, for maximization, the derivative of each of the i's must equal 0. Thus we can ignore the Σ. ∂lnW/∂Pi = -lnPi - 1 - α - βEi = 0 ∴ lnPi = -(1 + α) - βEi Or Pi = exp(-(1 + α))exp(-βEi) Write as, Pi = (1/Z)exp(-βEi) This is the BOLTZMANN DISTRIBUTION. The Partition Function ----------------------- From the discussion of the Boltzmann distribution it was shown that: Pi = (1/Z)exp(-βEi) Now impose the above constraints ΣPi = 1 and E = ΣiPiEi Thus, ΣPi = 1: (1/Z)Σie-βEi = 1 ∴ Z = Σie-βEi This is the PARTITION FUNCTION E = ΣiPiEi: Take the derivative of Z w.r.t. β ∂Z/∂β = -ΣiEiexp(-βEi) Divide both sides by -Z: (-1/Z)∂Z/∂β = ΣiEiexp(-βEi)/Z The RHS contains the Boltzmann distribution Pi = (1/Z)exp(-βEi) = ΣiPiEi = E ∴ E = (-1/Z)∂Z/∂β This can be written as: E = -∂lnZ/∂β Entropy Revisited ----------------- S = -ΣiPilnPi and Pi = (1/Z)exp(-βEi) Therefore S can be wriiten as: S = -Σi(1/Z)e-βEilnPi Now, lnPi = (-βEi - lnZ). Therefore, S = -Σi(1/Z)e-βEi(-βPi - lnZ) = Σi(1/Z)e-βEi(βEi + lnZ) = βΣi(1/Z)e-βEiEi + lnZ(1/Z)Σie-βEi = βΣiPiEi + lnZ(1/Z)z = βE + lnZ Now, dS = βdE + Edβ + (∂lnZ/∂β)dβ but ∂lnZ/∂β = -E so dS = βdE or, β = dS/dE Helmholtz Free Energy --------------------- Temperature is defined as: T = (1/kB)dE/dS The temperature, T, is the change in E that causes a change in the entropy by 1J/K. Alternatively, dS/dE = 1/KBT By comparison, β = 1/KBT Now from before S = βE + lnZ. Therefore, S = (1/KBT)E + lnZ Or E - KBTS = A = -KBTlnZ This is the HELMHOLTZ FREE ENERGY. Summarizing:
Pi = (1/Z)exp(-βEi) Z = Σie-βEi E = -∂lnZ/∂β KBT = 1/β S = βE + lnZ A = -KBTlnZ