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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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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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Double Slit Experiment

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Elastic and Inelastic Collisions .

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Electric Fields

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Gauss's Law of Universal Gravity .

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Gravity - Force and Acceleration

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Ideal and Non-Ideal Gas Laws (van der Waal)

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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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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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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 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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Cosmic Background Radiation and Decoupling

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Friedmann-Robertson-Walker Equations

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Geometries of the Universe

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Inflation Theory

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Introduction to Black Holes .

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Olbers' Paradox

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Stephen Hawking's Last Paper .

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Stephen Hawking's PhD Thesis .

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Finance and Accounting

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Brownian Model of Financial Markets

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Lecture Notes on International Financial Management

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Periodically and Continuously Compounded Interest

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Repurchase versus Dividend Analysis

Game Theory

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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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Gravitational Time Dilation

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Quantum Gravity

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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 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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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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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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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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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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Regular Polygons

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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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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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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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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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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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Survey Data - Likert Item and Scale

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Programming and Computer Science

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How this site works ...

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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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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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Clebsch-Gordan Coefficients .

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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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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 P-N Junction

Special Relativity

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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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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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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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Last modified: November 20, 2021 ✓


Kinetic Theory of Gases
-----------------------

An ideal gas is defined as one in which all 
collisions between atoms or molecules are 
perfectly elastic and in which there are no 
intermolecular attractive forces. One can 
visualize it as a collection of perfectly 
hard spheres which collide but which otherwise 
do not interact with each other. In such a gas, 
all the internal energy is in the form of 
kinetic energy and any change in internal 
energy is accompanied by a change in 
temperature.



Consider a molecule striking the wall of a box.  
Assume the collision with the wall is perfectly 
elastic so that the change in the x momentum is 
given by:

mv_x - (-mv_x) = 2mv_x

Now F_x = ma_x = 2mv_x/Δt

Now, if we assume the molecule makes a 
'round trip' we get: 

Δt = 2L/v_x

So,

F_x = 2mv_x/(2L/v_x) = mv_x²/L

The force due to N molecules is Nmv_x²/L

Now v² = v_x² + v_y² + v_z² = 3v_x²

∴ v_x² = v²/3

P = F/A =  Nmv²/3LA = Nmv²/3V

∴ PV = Nmv²/3 = nRT ... the Ideal Gas Law

Now,

n = N/N_A where N_A is Avogadro's Number

∴ Nmv²/3 = NRT/N_A

mv²/3 = RT/N_A

mv² = 3k_BT where k_B =  R/N_A = BOLTZMANN'S CONSTANT

∴ (1/2)mv² = 3k_BT/2 

The total INTERNAL ENERGY of the gas is thus:

U =  3Nk_BT/2 = 3nRT/2