Redshift Academy

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


Avogadro's Number
Balancing Chemical Equations
The Periodic Table

Classical Physics

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

Climate Change

Keeling Curve


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

Finance and Accounting

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

Game Theory

The Truel

General Relativity

Accelerated Reference Frames - Rindler Coordinates
Catalog of Spacetimes
Curvature and Parallel Transport
Dirac Equation in Curved Spacetime
Einstein's Field Equations
Gravitational Time Dilation
Gravitational Waves
Quantum Gravity
Relativistic, Cosmological and Gravitational Redshift
Ricci Decomposition
Ricci Flow
Stress-Energy Tensor
Stress-Energy-Momentum Tensor
The Area Metric
The Equivalence Principal
The Essential Mathematics of General Relativity
The Induced Metric
The Metric Tensor
Vierbein (Frame) Fields
World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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


Lecture Notes on International Economics
Lecture Notes on Macroeconomics
Macroeconomic Policy


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


Marginal Revenue and Cost

Particle Physics

Feynman Diagrams and Loops
Field Dimensions
Helicity and Chirality
Klein-Gordon and Dirac Equations
Regularization and Renormalization
Scattering - Mandelstam Variables
Spin 1 Eigenvectors
The Vacuum Catastrophe

Probability and Statistics

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

Programming and Computer Science

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

Quantum Computing

The Qubit

Quantum Field Theory

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

Quantum Mechanics

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

Semiconductor Reliability

The Weibull Distribution

Solid State Electronics

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

Special Relativity

Electromagnetic 4 - Potential
Energy and Momentum, E = mc2
Lorentz Invariance
Lorentz Transform
Lorentz Transformation of the EM Field
Newton versus Einstein
Spinors - Part 1
Spinors - Part 2
The Lorentz Group
Velocity Addition

Statistical Mechanics

Black Body Radiation
Entropy and the Partition Function
The Harmonic Oscillator
The Ideal Gas

String Theory

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


BCS Theory
Introduction to Superconductors
Superconductivity (Lectures 1 - 10)
Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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

The Standard Model

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


Units, Constants and Useful Formulas



Certain metals and alloys, when cooled to extremely low temperatures, suddenly lose all their electrical resistance and become superconductors. The temperature at which the transition to the superconducting state occurs is referred to as the critical temperature, Tc. Superconductors also exhibit remarkable magnetic behavior and they can have their superconductivity destroyed by the application of a sufficiently large magmetic field. For many purpsoses a superconductor can be viewed as consisting of two interpenetrating fluids. One of the fluids is the normal fluid and is composed of normal conduction electrons obeying Fermi-Dirac statistics and the other is the superfluid. According to the BCS theory, the superfluid consists of pairs of electrons (Cooper pairs) of equal but opposite momentum and spin. The pairing is between electrons which lie near the Fermi energy of the normal metal and arises because of a subtle interaction between these electrons and the lattice. The resulting pairs are in an energy state lower then the Fermi energy by an amount corresponding to the binding energy of the pair. It can be shown from quantum theory that it is energetically favourable for all the Cooper pairs to be in the same quantum state and have the same momentum. This is possible because a Cooper pair having integral (zero) spin can be thought of as a Bose particle. This means that the only scattering process which can reduce the current flow in a superconductor is one in which the pairs are broken up, a process which requires at least an energy equal to the binding energy of the pair. For low current desnsities there is not a way in which this energy can be imparted to the pairs and consequently the specimen has no resistance. As with ordinary particles it is possible to represent the motion of a Cooper pair by a wavefunction whose gradient determines the magnitude of the current flowing in the superconductor. This wavefunction has the form of a travelling wave whose phase coherence extends over indefinitely large distances. When superconductivity is destroyed, for example by a large current or magnetic field, this phase coherence is lost.