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Astronomy

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

Chemistry

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Avogadro's Number .
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table .

Classical Mechanics

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Blackbody Radiation .

Classical Physics

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

Climate Change

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Keeling Curve .

Cosmology

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

Finance and Accounting

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

Game Theory

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The Truel .

General Relativity

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Basis One-forms .
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Einstein's Field Equations
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Geodesics
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Gravitational Waves
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Hyperbolic Motion and Rindler Coordinates .
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Quantum Gravity
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Ricci Decomposition .
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Ricci Flow .
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Stress-Energy-Momentum Tensor .
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Tensors
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The Area Metric
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The Dirac Equation in Curved Spacetime .
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The Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Light Cone .
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The Metric Tensor .
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The Principle of Least Action in Relativity .
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Vierbein (Frame) Fields

Group Theory

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Basic Group Theory .
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Basic Representation Theory .
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Building Groups From Other Groups .
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Sets, Groups, Modules, Rings and Vector Spaces
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Symmetric Groups .
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The Integers Modulo n Under + and x .

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory .
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Euler-Lagrange Equation .
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics .
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Hamiltonian Formulation .
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem .

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series .
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Binomial Theorem (Pascal's Triangle)
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations .
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Conjugate Pair Theorem
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Contravariant and Covariant Components of a Vector
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Derivatives of Inverse Functions
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Double Angle Formulas
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Eigenvectors and Eigenvalues
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Euler Formula for Polyhedrons
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Factoring of a3 +/- b3
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Fourier Series and Transforms .
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Fractals
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Gauss's Divergence Theorem
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Grassmann and Clifford Algebras .
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Heron's Formula
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Index Notation (Tensors and Matrices) .
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Inequalities
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Integration By Parts
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Introduction to Conformal Field Theory .
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Inverse of a Function
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Law of Sines and Cosines
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Line Integrals, ∮
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Logarithms and Logarithmic Equations
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Matrices and Determinants
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Matrix Exponential
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Mean Value and Rolle's Theorem
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Modulus Equations
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Orthogonal Curvilinear Coordinates .
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Parabolas, Ellipses and Hyperbolas
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Piecewise Functions
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Polar Coordinates
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Polynomial Division
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Quaternions 1 .
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Quaternions 2 .
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Regular Polygons
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Related Rates
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Similar Matrices and Diagonalization .
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Spherical Trigonometry
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Stirling's Approximation
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Sum and Differences of Squares and Cubes
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Symbolic Logic
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Tangent and Normal Line
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Taylor and Maclaurin Series .
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The Essential Mathematics of Lie Groups
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The Limit Definition of the Exponential Function
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Nuclear Physics

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Radioactive Decay

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity, Chirality and Weyl Spinors .
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Klein-Gordon and Dirac Equations .
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors .

Probability and Statistics

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Box and Whisker Plots
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Buffon's Needle .
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Computing

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Density Operators and Mixed States .
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Entangled States .
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The Qubit .

Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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Bohr Atom
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Clebsch-Gordan Coefficients .
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Commutators
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Dyson Series
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Electron Orbital Angular Momentum and Spin
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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions .
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Pauli Spin Matrices
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Photoelectric Effect .
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Position and Momentum States .
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Probability Current
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Schrodinger Equation for Hydrogen Atom .
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Schrodinger Wave Equation
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Quantum Harmonic Oscillator .
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .
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Fermi-Dirac Statistics .
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Intrinsic and Extrinsic Semiconductors .
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors .
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Electromagnetic (Faraday) Tensor .
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Energy and Momentum in Special Relativity, E = mc2 .
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Invariance of the Velocity of Light .
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Lorentz Invariance .
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Lorentz Transform .
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Lorentz Transformation of the EM Field .
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Newton versus Einstein
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Spinors - Part 1 .
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Spinors - Part 2 .
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The Continuity Equation .
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The Lorentz Group .

Statistical Mechanics

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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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Bardeen–Cooper–Schrieffer Theory
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BCS Theory
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Cooper Pairs
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Introduction to Superconductivity .
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1 .
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3 .
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
Last modified: March 2, 2022 ✓

Vector Calculus Primer ---------------------- Let φ be a scalar and F be a vector, F = <P,Q,R) ∇ = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k = (∂/∂x,∂/∂y,∂/∂z) = operator Gradient of φ = ∇φ = (∂φ/∂x)i + (∂φ/∂y)j + (∂φ/∂z)k = a vector Curl of F = ∇ x F = (∂R/∂y - ∂Q/∂z)i + (∂P/∂z - ∂R/∂x)j + (∂Q/∂x - ∂P/∂y)k | i j k | = | ∂/∂x ∂/∂y ∂/∂z | | P Q R | = a vector The curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space. Divergence of F = ∇.F - - - - = | ∂/∂x ∂/∂y ∂/∂z || P | - - | Q | | R | - - = ∂P/∂x + ∂Q/∂y + ∂R/∂z = a scalar The divergence measures how much a vector field spreads out or diverges from a given point. Curl of the gradient of φ: ∇ x (∇φ) = 0 Proof: | i j k | | ∂/∂x ∂/∂y ∂/∂z | | ∂φ/∂x ∂φ/∂y ∂φ/∂z | = (∂2φ/∂y∂z - ∂2φ/∂z∂y)i + (∂2φ/∂x∂z - ∂2φ/∂z∂x)j + (∂2φ/∂x∂y - ∂2φ/∂y∂x)k = 0 This means the gradient of a scalar field has no rotation. Divergence of the curl of F: ∇.(∇ x F) = 0 Proof: - - - - | ∂/∂x ∂/∂y ∂/∂z || ∂R/∂y - ∂Q/∂z | - - | ∂P/∂z - ∂R/∂x | | ∂Q/∂x - ∂P/∂y | - - = (∂/∂x)(∂R/∂y - ∂Q/∂z) + (∂/∂y)(∂P/∂z - ∂R/∂x)j + (∂/∂z)(∂Q/∂x - ∂P/∂y)k = ∂2R/∂x∂y - ∂2Q/∂x∂z + ∂2P/∂y∂z - ∂2R/∂y∂x + ∂2Q/∂z∂x - ∂2P/∂z∂y = 0 Divergence of the gradient of φ: ∇.(∇φ) = ∇2φ Proof: - - - - | ∂/∂x ∂/∂y ∂/∂z || ∂φ/∂x | - - | ∂φ/∂y | | ∂φ/∂z | - - = (∂2φ/∂x) + (∂2φ/∂y) + (∂2φ/∂z) = a scalar Curl of the curl of F: ∇ x (∇ x F) = ∇(∇.F) - ∇2F Proof | i j k | | ∂/∂x ∂/∂y ∂/∂z | | ∂R/∂y - ∂Q/∂z ∂P/∂z - ∂R/∂x ∂Q/∂x - ∂P/∂y | = [∂2Q/∂y∂x + ∂2R/∂z∂x - ∂2P/∂y2 - ∂2P/∂z2 - ∂2P/∂x2 + ∂2P/∂x2]i ---------------- Dummy term = [∂2Q/∂y∂x + ∂2R/∂z∂x + ∂2P/∂x2 - (∂2P/∂y2 + ∂2P/∂z2 + ∂2P/∂x2)]i = [(∂/∂x)(∂Q/∂y + ∂R/∂z + ∂P/∂x) - (∂2P/∂y2 + ∂2P/∂z2 + ∂2P/∂x2)]i = [(∂/∂x)(∂Q/∂y + ∂R/∂z + ∂P/∂x) - (∂2P/∂y2 + ∂2P/∂z2 + ∂2P/∂x2)]i + [(∂/∂y)(∂Q/∂y + ∂R/∂z + ∂P/∂x) - (∂2Q/∂y2 + ∂2Q/∂z2 + ∂2Q/∂x2)]j + [(∂/∂z)(∂Q/∂y + ∂R/∂z + ∂P/∂x) - (∂2R/∂y2 + ∂2R/∂z2 + ∂2R/∂x2)]k Now, (∂/∂x)i + (∂/∂y)j + (∂/∂z)k = ∇ and, - - - - (∂Q/∂y + ∂R/∂z + ∂P/∂x) = | ∂/∂x ∂/∂y ∂/∂z || P | - - | Q | | R | - - = ∇.F and, [∂2P/∂y2 + ∂2P/∂z2 + ∂2P/∂x2]i + [...]j + [...]k = ∇2F Therefore, ∇ x (∇ x F) = ∇(∇.F) - ∇2φ Q.E.D Lorentz Force Law ----------------- In general, the force is defined as the sum of the electric force and the magnetic force as: F = q(E + v x B) where v is the velocity = q(E + vBsinθ) where θ is the angle between v and B = q(E + {Fx + Fy + Fz}) = q(E + {(vyBz - vzBy) + (vzBx - vxBz) + (vxBy - vyBx)}) ++++++++ ---------- + ^ E Field: |E | |F The direction of the force v v - is parallel to the field. ---------- -------- N ---------- q.---->v |B B Field: v The direction of the force F is determined by the Right ---------- Hand Rule. S Right Hand Rule --------------- F | /B | / |/ q -----v Place thumb in direction of v, fingers in direction of B, palm in direction of F for +ve charge. Do the opposite for -ve charge. The RHR is implicit in the component representation of the cross product. Thus, for a +ve charge, B in the z direction and v in the x direction we get: F = q{(vyBz - vzBy) + (vzBx - vxBz) + (vxBy - vyBx)} => Fy = -qvxBz Which is in accordance with the RHR. Maxwell's Equations ------------------- Maxwell’s equations explain how the electric charges and electric currents produce magnetic and electric fields. The equations describe how the electric field can create a magnetic field and vice versa. They are the set of 4 partial differential equations, along with the Lorentz force law, that form the foundation of classical electrodynamics, electric circuits and classical optics. The equations can be written in both integral and differential form. Gauss's Law for Electricity --------------------------- Maxwell first equation is based on the Gauss's law which states that the closed surface integral of electric flux density is always equal to charge enclosed by that surface. ∯E.dS = Q/ε0 Q = ∭ρdV ∯E.dS = (1/ε0)∭ρdV S V Now, ∭∇.EdV = ∯E.dS (Divergence theorem) V S = (1/ε0)∭ρdV V Therefore, ∭(∇.E - ρ/ε)dV = 0 V V cannot be 0. Therefore: ∇.E - ρ/ε = 0 Gauss's Law for Magnetism ------------------------- Maxwell's second equation is based on Gauss's law of magnetism which states that the net magnetic flux out of any closed surface is zero. This is the statement that magnetic monopole sources that are analogous to charge in the electric field case, cannot exist. For a magnetic dipole, the magnetic flux directed inward toward the south pole will equal the flux outward from the north pole. Therefore, the net flux will always be zero for dipole sources. From the electric case: ∯E.dS = ρ/ε0 S If we replace E with B and set ρ = 0 (no magnetic charge (monopoles)), we get: ∯B.dS = 0 S Now, ∭∇.BdV = ∯B.dS (Divergence theorem) V S = 0 V cannot be 0. Therefore: ∇.B = 0 Faraday's law of Induction (Maxwell–Faraday equation) ----------------------------------------------------- Maxwell’s 3rd equation is derived from Faraday’s laws of Electromagnetic Induction. The magnetic flux, φB = BS where S is an area perpendicular to B. This can be put in integral form as: φB = ∯B.dS S The EMF, V, is given by: V = -dφB/dt EMF in a circuit is the net work done by the driving force per unit charge in the circuit. The driving force in a circuit must come from electric field since magnetic force on a moving charge is perpendicular to velocity, and hence, can do no work. This is a consequence of the Lorentz force law, F = q(E + v x B). Since electric field is equal to the force per unit charge, the induced EMF will be equal to the line integral of the electric field around the circuit. The line integral calculated in the direction of the electric field gives a positive value for the EMF. Therefore, V = Work done = force x dist = ∮E.dl C Therefore, ∮E.dl = -dφB/dt = -d/dt{∯B.dS} C S = -∯(∂B/∂t).dS) S Now, ∮E.dl = ∯(∇ x E).dS (Stoke's theorem) C S Therefore, -∯(∂B/∂t).dS) = ∯(∇ x E).dS S S or, ∯(∇ x E + ∂B/∂t).dS = 0 S S cannot be 0. Therefore: ∇ x E + ∂B/∂t = 0 Ampere's Law with Maxwell Correction ------------------------------------ Maxwell's 4th equation is based on Ampere’s circuit law. Ampere's law states that the magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source. Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop. ∮B.dl = μ0I C Now, I = ∯J.dS S Therefore, ∮B.dl = ∯μ0J.dS C S ∮B.dl = ∯(∇ x B).dS (Stoke's theorem) C S ∯μ0J.dS = ∯(∇ x B).dS S S or, ∯(∇ x B - μ0j).dS = 0 S S cannot be 0. Therefore: ∇ x B - μ0j = 0 With Maxwell's correction this becomes: ∇ x B - μ0(j + ε0∂E/∂t) = 0 The displacement current correction was added to Ampere's law by Maxwell to explain the presence of a magnetic field between capacitor plates even though no current is threading through the path. Summary ------- The 4 Maxwell equations in integral form are: ∫E.dA = Q/ε0 ∫B.dA = 0 Ε = ∫E.dl = -∂φB/∂t ∫B.dl = μ0I + μ0ε0∂φE/∂t The correspnding equations in differential form are: ∇.E = ρ/ε0 ∇.B = 0 ∇ x E = -∂B/∂t ∇ x B = μ0J + μ0ε0∂E/∂t Both the differential and integral formulations are useful. The integral formulation can often be used to simply and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential formulation is a more natural starting point for calculating the fields in more complicated (less symmetric) situations. Electromagnetic Wave Equation in Free Space ------------------------------------------- To obtain the electromagnetic wave equation in a free space vacuum we set ρ = J = 0. This gives: ∇.E = 0 ∇.B = 0 ∇ x E = -∂B/∂t ... 1. ∇ x B = μ0ε0∂E/∂t ... 2. From 1. ∇ x (∇ x E) = -∂(∇ x B)/∂t Substitute 2 into 1: ∇ x (∇ x E) = (-∂/∂t)(μ0J + μ0ε0∂E/∂t) = -μ0ε02E/∂t2 Use the identity ∇ x (∇ x E) = -∇2E + ∇(∇.E) But ∇.E = 0. Thus, ∇2E = μ0ε02E/∂t2 One solution is E = E0sin{2π(x - vt)/λ} check: ∇2E = -α2sin{α(x - vt)} where α = 2π/λ μ0ε02E/∂t2 = -μ0ε0α2v2sin{α(x - vt)} From which we get: v2 = 1/μ0ε0 = c2 Therefore, ∇2E = (1/c2)∂2E/∂t2 Similarly, from 2. ∇ x (∇ x B) = ∇ x μ0ε0∂E/∂t = ∇ x μ0ε0∂E/∂t = μ0ε0∂(∇ x E)/∂t = μ0ε0∂(-∂B/∂t)/∂t using 1. ∇(∇.B) - ∇2B = μ0ε0∂(-∂B/∂t)/∂t But ∇.B = 0 so: -∇2B = μ0ε0∂(-∂B/∂t)/∂t or, ∇2B = μ0ε02B/∂t2 This has a solution similar to E: B = B0sin{2π(x - vt)/λ} Light is an self sustaining e-m wave! Changing E field => changing B field => changing E field. The B field is perpendicular to the E field and both have the same phase (i.e. zero phase difference). y | --> propagation | | E | | | | | E d|...........|c E+dE | : B : dy | a|...........|b o-------+-----------+--------> / /x / x+dx / / / / / /B+dB z B / https://www.youtube.com/watch?v=8YkHEtq0bhc b c d a ∮E.dl + ∮E.dl + ∮E.dl + ∮E.dl = -dφB/dt a b c d 0 + (E + dE)dy + 0 - Edy = -d(Bdxdy)/dt dE = -d(Bdx)/dt dE/dx = -dB/dt Therefore, dE0sin{2π(x - vt)/λ}/dx = -dB0sin{2π(x - vt)/λ}/dt E0(2π/λ)cos{2π(x - vt)/λ} = B0(2πv/λ)cos{2π(x - vt)/λ} ∴ E0 = B0v or, v = c = E0/B0 Thus, the ratio of the electric to magnetic fields in an electromagnetic wave in free space is always equal to the speed of light. Energy density of E field: UE = ε0E2/2 J/m3 This is derived from the energy stored in a capacitor. Energy density of B field: UB = B2/2μo J/m3 This is derived from the energy stored in a solenoid. UTotal = ε0E2/2 + B2/2μo Now B = E/c so: UB = E2/2μ0c2 Now, c2 = 1/ε0μ0 so UB = E2ε0μ0/2μ0 = E2/2ε0 Thus, we can write UTotal in the following equivalent forms: UTotal = ε0E2 ≡ B2o ... substituting E = cB in the UE equation. ≡ cε0EB The intensity can be found by taking the energy density (energy per unit volume) at a point in space and multiplying it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area. I0 = cU0 = cε0E02/2 + cB02/2μo = cε0E02 ≡ cB02o ≡ c2ε0E0B0 Now, c2 = 1/(μ0ε0) Therefore, I0 = {1/(μ0ε0)}{ε0E0B0} = (1/μ0)EB This is the POYNTING VECTOR. This more formally defined as: - S = (1/μ0)ExB = (1/μ0)EBsinθ = (1/μ0)EB since E and B are orthogonal The Poynting vector represents the rate of energy transport per unit area (energy flux) in W per m2). The modulus of S, |S| is equal to the intensity, I. All electromagnetic waves (radio, light, X-rays, etc.) obey the inverse-square law thus the intensity of an electromagnetic wave is proportional to the inverse of the square of the distance from a point source. ERMS = E0/√2 and BRMS = B0/√2 IAverage = cε0ERMS2 = cBRMS2o Photons ------- In the quantum description, the electromagnetic field is an observable property of photons. Photons can be thought of as mini E/M wave segments that consists of an oscillating electric field component, E, and an oscillating magnetic field component, B. The electric and magnetic fields are orthogonal (perpendicular) to each other, and they are orthogonal to the direction of propogation of the photon. The E and B fields flip direction as the photon travels. The number of oscillations that occur in one second is the frequency, f. The superposition of a sufficiently large number of photons has the characteristics of a continuous electromagnetic wave. The energy of the photon is given by E = hf and a wavelength is equal to c/f. There is a correspondence between the energy of a photon stream and the Poynting vector in the classical approach. Laplace's and Poisson's Equations --------------------------------- Maxwell's equations again: ∇.E = -ρ/ε0 ∇.B = 0 ∇ x E = -∂B/∂t ∇ x B = μ0J + μ0ε0∂E/∂t Static Case (Electrostatics) ---------------------------- Electric Field: ∇ x E = 0 Using the identity that ∇ x (-∇φ) = 0 we can imply that, E = -∇φ where φ is the ELECTRIC or SCALAR POTENTIAL Therefore, ∇.E = ρ/ε0 becomes ∇.(∇φ) = -ρ/ε02φ = -ρ/ε0 This is POISSON'S's equation. In a charge free space: ∇2φ = 0 This is LAPLACE's equation. Magnetic Field: ∇.B = 0 Using the identity that ∇.(∇ x A) = 0 we can imply that B = ∇ x A where we define A as the MAGNETIC or VECTOR POTENTIAL. ∇ x B = μ0J (static case: μ0ε0∂E/∂t = 0) ∴ ∇ x (∇ x A) = μ0J ∴ ∇(∇.A) - ∇2A = μ0J We can simplify this equation by 'fixing' the gauge. If we set ∇.A = 0 (the Coulomb gauge) we get: ∇2A = -μ0J Dynamic Case (Electrodynamics) ------------------------------ Electric Field: ∇ x E = -∂B/∂t = -∂(∇ x A)/∂t ∇ x E + ∂(∇ x A)/∂t = 0 Which can be written as, ∇ x (E + ∂A/∂t) = 0 Using the identity that ∇ x (∇φ) = 0 we can imply that: ∇φ = (E + ∂A/∂t) Therefore, E = -∇φ - ∂A/∂t or ∇φ = -E - ∂A/∂t ∇.(∇φ) = -∇.E - ∇.(∂A/∂t) ∇2φ = -ρ/ε0 - ∂(∇.A)/∂t Magnetic Field: ∇ x B = μ0J + μ0ε0∂E/∂t ∴ ∇ x (∇ x A) = μ0J + μ0ε0∂(-∇φ - ∂A/∂t)/∂t ∴ -∇2A = -∇(∇.A) + μ0J - μ0ε0∇(∂φ/∂t) - μ0ε02A/∂t2 ∴ -∇2A = μ0J - ∇[∇.A + μ0ε0(∂φ/∂t)] - μ0ε02A/∂t2 ∴ ∇2A = -μ0J + ∇[∇.A + μ0ε0(∂φ/∂t)] + μ0ε02A/∂t2 We can simplify this equation by 'fixing' the gauge. If we set ∇.A + μ0ε0∂φ/∂t = 0 we get: ∇2A - μ0ε02A/∂t2 = -μ0J We can also apply the same gauge condition to, ∇2φ = -ρ/ε0 - ∂(∇.A)/∂t Which was derived above. We get: ∇2φ = -ρ/ε0 - ∂(-μ0ε0∂φ/∂t)/∂t (∇.A = -μ0ε0∂φ/∂t) ∴ ∇2φ = -ρ/ε0 + μ0ε02φ/∂t2 In summary, ∇2A - μ0ε02A/∂t2 = -μ0J and ∇2φ - μ0ε02φ/∂t2 = -ρ/ε0 These are Maxwell's equations in terms of the scalar and vector potentials. φ and A make up 4 functions in total (1 for the scalar function and 1 for each component of A). This is a simplification over the original equations, which make up 6 functions (3 for each component of E and B). Notice that they reduce to the familiar static equations when A and φ do not depend on time. Gauge Transformations --------------------- The gauge referred to above is called the LORENZ GAUGE (not to be confused with Lorentz). ∇.A + μ0ε0∂φ/∂t = 0 ∇.A' + μ0ε0∂φ'/∂t = ∇.A + ∇2λ + μ0ε0∂φ/∂t - μ0ε02λ/∂t2 = 0 ∴ ∇2λ - μ0ε02λ/∂t2 = ∇.A + μ0ε0∂φ/∂t So as long as we choose λ that meets the requirement that, ∇2λ - μ0ε02λ/∂t2 = 0 the Lorenz requirement will be satisfied. The Lorenz gauge is very useful in electrodynamics because of the simple relations it leads to for A and φ. There is another gauge that simplifies relations in electrostatics. This is called the COULOMB GAUGE. Consider: A' = A + ∇λ ∴ ∇ x A' = ∇ x A + ∇ x (∇λ) = ∇ x A since ∇ x (∇λ) = 0 ∇.A' = ∇.A + ∇.(∇λ) = ∇.A + ∇2λ If λ is chosen such that ∇2λ = -∇.A then ∇.A' = 0.