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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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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Lorentz Force Law

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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 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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Penrose Diagrams

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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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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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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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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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Stress-Energy Tensor

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Stress-Energy-Momentum Tensor

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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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Augmented Matrices and Cramer's Rule

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Basic Group Theory

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Basic Representation Theory

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Binomial Theorem (Pascal's Triangle)

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Building Groups From Other Groups

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Completing the Square

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Complex Numbers

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Composite Functions

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Conformal Transformations .

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Conjugate Pair Theorem

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Contravariant and Covariant Components of a Vector

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Derivatives of Inverse Functions

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Double Angle Formulas

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Eigenvectors and Eigenvalues

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Euler Formula for Polyhedrons

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Factoring of a³ +/- b³

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Fourier Series and Transforms .

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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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Stem Plots, Histograms and Ogives

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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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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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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: January 26, 2018


Polar Coordinates
-----------------

                         P
                  ^ y   (x,y)           P = (r,θ) 
                  |    /                  ≡ (r,θ + 2nπ)
                  |   /                 x = rcosθ
                  |  /r                 y = rsinθ
                  | /
                  |/θ
    --------------+------------------>x
                 /|  
                / |
               /  |                    
              /   |
             Q    |   Q = (r,θ + π) 
          (-x,-y)       ≡ (-r,θ)
                        ≡ (-r,θ + (2n + 1)π)

Slope of Tangent Line at θ
---------------------------

        dy/dθ
dy/dx = -----  ... CHAIN RULE
        dx/dθ


Horizontal tangent line:  dy/dθ = 0

Vertical tangent line:  dx/dθ = 0

Example:

r = 2sinθ

x = rcosθ = 2sinθcosθ

y = rsinθ = 2sin²θ

dx/dθ = 2cos²θ - 2sin²θ = 0

∴ θ = π/4, 3π/4, 5π/4, 7π/4

dy/dθ = 4sinθcosθ = 0

∴ θ = 0, π/2, π, 3π/2

Not all of these are θs lines correspond to tangent 
lines.  To determine the valid ones it is necessary 
to look at the polar plot.



We can now see that:

π/4 and 3π/4 for vertical tangent lines.

0 and π/2 for horizontal tangent lines.

The r coordinate is found by solving the original polar
equation.  Therefore,

r = 2sin(π/4) = √3

r = 2sin(3π/4) = √3

r = 2sin(0) = 0

r = 2sin(π/2) = 2

Therefore, the polar coordinates of the tangent lines 
are:

(√3,π/4) (√3,3π/4) (0,0) (2,π/2)