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Astronomy

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Celestial Coordinates
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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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Classical Physics

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Archimedes Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Comparison Between Gravitation and Electrostatics
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Compton Effect
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Elastic and Inelastic Collisions
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Refractive Index
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Rotational Dynamics
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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 Zeeman Effect
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Wien's Displacement Law
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Young's Modulus

Climate Change

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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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Geometries of the Universe
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Inflation Theory
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Introduction to Black Holes
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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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Annuities
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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

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Einstein's Field Equations
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Quantum Gravity
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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 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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Symmetry and Conservation Laws - Noether's Theorem

Macroeconomics

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Mathematics

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Amplitude, Period and Phase
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Basic Group Theory
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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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Contravariant and Covariant Components of a Vector
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Grassmann and Clifford Algebras
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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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Orthogonal Curvilinear Coordinates
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Quaternions 1
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Quaternions 2
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Regular Polygons
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Symmetric Groups
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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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Programming and Computer Science

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Hashing
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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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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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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 (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: July 30, 2018

Quaternions ----------- Quaternions are a number system that extends the complex numbers. They are represented in the form: q = a + bi + cj + dk Where i2 = j2 = k2 = ijk = -1 They are cyclic, i.e. i ij = k ji = -k ^ \ jk = i ik = -j / v ki = j etc. k <- j Proof: ijk = -1 iijk = -i -jk = -i jk = 1 Quaternions can be decomposed into scalar and vector parts: q = (a,b,c,d) = (a,v) ^ ----- | ^ scalar \ vector A quaternion (0,v) is called a PURE quaternion. A UNIT quaternion is a quaternion of modulus one. (q = q/√(a2 + b2 + c2 + d2) The Quaternion Group, Q8 ------------------------ Quaternions form a group, Q8 = {1,-1,i,-i,j,-j,k,-k} Operations: + or . Closure: q1 + q2/ = q3 q1.q2 = q3 Inverse: q1 + (-q1) = 0 q1.q1* = 1 Identity: 0 + q1 = q1 1.q1 = q1 Associativity: q1 + (q2 + q3) = (q1 + q2) + q3) (q1q2)q3 = q1(q2)q3q2) Commutativity: q1 + q2 = q2 + q1 q1.q2 ≠ q2.q1 [i,j] = ij - ji = k - (-k) = 2k Subgroups: {1,-1,i,-i} {1,-1,j,-j} {1,-1,k,-k} {1,-1} = center (kernel) {1} Quaternion Multiplication ------------------------- q1 = (a,b,c,d) q2 = (e,f,g,h) q1q2 = (ae - bf - cg - dh, af + be + ch - dg, ag - bh + ce + df, ah + bg - cf + de) This can also be written in terms of real matrices as follows: - - - - | a -b -c -d || e | | b a -d c || f | | c d a -b || g | | d -c b a || h | - - - - Example: (1,2,3,6)(0,1,0,0) - - - - - - | 1 -2 -3 -6 || 0 | | -2 | | 2 1 -6 3 || 1 | = | 1 | | 3 6 1 -2 || 0 | | 6 | | 6 -3 2 1 || 0 | | -3 | - - - - - - Individual elements are given by: - - | 1 0 0 0 | (1,0,0,0) = | 0 1 0 0 | | 0 0 1 0 | | 0 0 0 1 | - - - - | 0 -1 0 0 | (0,1,0,0) = | 1 0 0 0 | | 0 0 0 -1 | | d 0 1 0 | - - etc. Extracting the Dot and Cross Products ------------------------------------- Let q1 = (a,x) and q2 = (b,y) q1q2 = (ae - (bf + cg + dh), af + be + ch - dg, ag - bh + ce + df, ah + bg - cf + de) (ae - x.y,ay + ex + x ^ y) Vector Rotation (Perpendicular Axis) ------------------------------------ n ^ v ^ | | v' ☉ means out of screen | / | / | / |/θ n ☉-----------> v v' = vcosθ + (n ^ v)sinθ Quaternion Rotation (Perpendicular Axis) ---------------------------------------- q = (0,v) q is the PURE quaternion corresponds to the vector being rotated. q' = (0,v') qn = (0,n) n is the unit vector coming out of the screen corresponding to the axis of rotation, and qn is the correponding pure quaternion. We need to figure out what n ^ v equals. Try, qnq = (0,n)(0,v) = (0 - n.v, 0v + 0n + n ^ v) = (0,n ^ v) = n ^ v Therefore, analagous to the vector forn, we can write: q' = qcosθ + (qnq)sinθ = (cosθ + qnsinθ)q Now, qn2 = (0,n)(o,n) = (0 - n.n,n ^ n) = (-|n|2,0) = (-1,0) = -1 So qn is analagous to i so we can write the quaternion form of Euler's equation as: exp(qnθ) = (cosθ + qnsinθ) ... 1. and, q' = exp(qnθ)q Example: q = (0,0,1) = k qn = (0,1,0) = j θ = π/2 z | ^ q | | -------->------- y / qn / v q' / x q' = exp(θqn)q = exp(πj/2)k = jk = i = (1,0,0) Vector Rotation (Arbitrary Axis) -------------------------------- v = v + v v' = v' + v' = v + v' = v + cosθv + sinθ(n ^ v) = v + cosθ(v - v) + sinθ(n ^ v) n = nxi + nyj + nzk is a unit vector corresponding to the arbitrary axis of rotation. (n ^ v) = n ^ (v - v) = n ^ v - n ^ v = n ^ v    v' = (1 - cosθ)v + cosθv + sinθ(n ^ v) Now, v = (v.n)n projection of v onto n v' = (1 - cosθ)(v.n)n + cosθv + sinθ(n ^ v) This is the RODRIGUES ROTATION FORMULA. Example: n = (0,0,1) v = (1,0,0) θ = π/2 v' = (1 - cosθ)(v.n)n + cosθv + sinθ(n ^ v) = 0 + 0 + (0,0,1) ^ (1,0,0) = (0,1,0) Quaternion Rotation (Arbitrary Axis) ------------------------------------ q = (0,v) q = (0,v) q = (0,v) q' = (0,v') qn = (0,n) Where, qn = (0,nxi,nyj,nzk) is now a pure quaternion corresponding to the arbitrary axis of rotation, n. As before, q' = q + q' = q + exp(θ.qn)q = q + (cosθ + sinθnxi + sinθnyj + sinθnkk)q Now exp(θ.qn)q = qexp(-θ.qn) Proof: (cosθ,sinθn)(0,v) = (0,v)(cosθ,-sinθn) (0,cosθv + sinθ(n ^ v)) = (0,cosθv - sinθ(v ^ n)) (0,cosθv + sinθ(n ^ v)) = (0,cosθv + sinθ(n ^ v)) Q.E.D. Likewise, exp(θ.qn)q = qexp(θ.qn) Proof: [q1,q2] = 2(v1 ^ v2) q' = exp(θqn/2)exp(-θqn/2)q + exp(θqn/2)exp(θqn/2)q Using exp(-θqn/2)q = qexp(-θqn/2) and exp(θ.qn)q = qexp(-θ.qn) from above gives: q' = exp(θqn/2)qexp(-θqn/2) + exp(θqn/2)qexp(-θqn/2) = exp(θqn/2)(q + q)exp(-θqn/2) = exp(θqn/2)qexp(-θqn/2) In terms of sin and cos this is: g = exp(θqn/2) = cos(θ/2) + sin(θ/2)nxi + sin(θ/2)nyj + sin(θ/2)nkk g* = exp(-θqn/2) = cos(θ/2) - sin(θ/2)nxi - sin(θ/2)nyj - sin(θ/2)nkk Therefore, we can write: q' = gqg* = gqg-1 Example: qn = (0,0,0,1) q = (0,1,0,0) θ = π/2 g = cos(π/4) + sin(π/4)nzk = √2/2 + (√2/2)k = (1 + k)/√2 q' = gqg* = [(1,0,0,1)/√2](0,1,0,0)[(1,0,0,-1)/√2] = (0,0,1,0) Why the conjugate? Consider: q' = gq = [(1,0,0,1)/√2](0,1,0,0) = (0,1/√2,1/√2,0) This is a rotation of 45°. Therefore, the product gq does not accomplish the full rotation. by itself. Quaternion Rotation Matrix -------------------------- A quaternion rotation: q' = gqg* = gqg-1 With q = w + xi + yj + zk and qq* = 1 = w2 + x2 + y2 + z2 Can be algebraically manipulated (not proven here) into a matrix rotation q' = Rq, where R is the rotation matrix given by: - - | 1 - 2y2 - 2z2 2xy - 2zw 2xz + 2yw | R = | 2xy + 2zw 1 - 2x2 - 2z2 2yz - 2xw | | 2xz - 2yw 2yz + 2xw 1 - 2x2 - 2y2 | - - This is a rotation around the vector n = (x,y,z) (or equivalently qn = nxi + nyj + nzk) by an angle 2θ. We will demonstrate this using the previous example where qn = (0,0,0,1) q = (0,1,0,0) θ = π/2 ∴ 2θ = π. Plugging the numbers in gives: - - - - - - | -1 0 0 || 1 | | -1 | | 0 -1 0 || 0 | = | 0 | | 0 0 1 || 0 | | 0 | - - - - - - This is indeed a rotation of π! The 2:1 nature is apparent since both qn and -qn map to the same R, i.e. the z2 is always positive. Relation Between SU(2) and SO(3) -------------------------------- Instead of the 4 x 4 real matrices shown before, we can also write quaternions as 2 x 2 complex matrices as follows: q = a + bi +cj + dk = (a + bi) + (c + di)j ij = k = z + wj We can then write this as a matrix in the same way that we can write a complex number as a matrix. - - - - | z -w | = | a + bi -(c + di) | | w* z* | | c + di a - bi | - - - - ----------------------------------------------------- Digression: Complex Numbers as Matrices --------------------------- Consider the 2 complex numbers. (a + bi)(c + di) = ac - bd + (ad + bc)i This can be written as the matrix: - - - - | a -b || c | | b a || d | - - - - The matrix on the LHS enables us to write. - - - - 1 = | 1 0 | and i = | 0 -1 | | 0 1 | | 1 0 | - - - - Euler's formula can then be written as: exp(iθ) = cosθ + isinθ - - - - exp(| 0 -θ |) = | cosθ -sinθ | | θ 0 | | sinθ cosθ | - - - - Differentiating both sides w.r.t. θ gives: iexp(iθ) = -sinθ + icosθ - - - - - - | 0 -1 || cosθ -sinθ | = | -sinθ -cosθ | | 1 0 || sinθ cosθ | | cosθ -sinθ | - - - - - - ----------------------------------------------------- Example: (1,2,3,6)(0,1,0,0) - - - - | (1 + 2i) -(3 + 6i) || (0 + i) -(0 + 0i) | | (3 - 6i) (1 - 2i) || (0 - 0i) (0 - i) | - - - - - - = | (-2 + i) -(6 - 3i) | | (6 + 3i) (-2 - i) | - - - - | -2 | = | 1 | | 6 | | -3 | - - Which is exactly the same result from before. The following matrices satisfy these conditions: - - - - - - - - I = | 1 0 | i = | 0 -1 | j = | 0 i | k = | -i 0 | | 0 1 | | 1 0 | | i 0 | | 0 i | - - - - - - - - = -iσ2 = -iσ1 = -iσ3 It is easy to check that the matrices satisfy the quaternion identites. In addition, all have determinnt = 1 and i = -i, j = -j, k = -k. These are the Pauli matrices of SU(2) that are isomorphic to the quaternions of unit norm. Given the 2:1 correspndence between rotations of R and qqg-1, it implies that there is a 2:1 homomorphism from SU(2) to SO(3). This has very important consequences in the physics of spin. The unit quaternions can also be thought of as the group corresponding to the 3-sphere, S3, that gives the group Spin(3), which is isomorphic to SU(2) and also the double cover of SO(3).