Redshift Academy

Wolfram Alpha:

Search by keyword:

Astronomy

-

Astronomical Distance Units .

-

Celestial Coordinates .

-

Celestial Navigation .

-

Location of North and South Celestial Poles .

Chemistry

-

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

-

-

-

Double Slit Experiment

-

Elastic and Inelastic Collisions .

-

Electric Fields

-

-

-

-

Gauss's Law of Universal Gravity .

-

Gravity - Force and Acceleration

-

-

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

-

-

-

-

-

-

Kinetic Theory of Gases .

-

Kirchoff's Laws

-

Laplace's and Poisson's Equations

-

Lorentz Force Law

-

Maxwell's Equations

-

Moments and Torque

-

-

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 Laws of Thermodynamics

-

The Zeeman Effect .

-

Wien's Displacement Law

-

Young's Modulus

Climate Change

-

Keeling Curve .

Cosmology

-

-

Cosmic Background Radiation and Decoupling

-

-

-

Friedmann-Robertson-Walker Equations

-

Geometries of the Universe

-

-

Inflation Theory

-

Introduction to Black Holes .

-

Olbers' Paradox

-

Penrose Diagrams

-

-

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

-

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

-

-

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

Macroeconomics

-

Lecture Notes on International Economics

-

Lecture Notes on Macroeconomics

-

Macroeconomic Policy

Mathematics

-

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

-

-

-

Regular Polygons

-

-

Sets, Groups, Modules, Rings and Vector Spaces

-

Similar Matrices and Diagonalization .

-

Spherical Trigonometry

-

Stirling's Approximation

-

Sum and Differences of Squares and Cubes

-

-

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

-

-

Vector Calculus

-

Volume Integrals

Microeconomics

-

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

-

-

-

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

-

-

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

-

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

-

-

-

Clebsch-Gordan Coefficients .

-

-

-

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

-

String Theory

-

Bosonic Strings

-

Extra Dimensions

-

Introduction to String Theory

-

Kaluza-Klein Compactification of Closed Strings

-

Strings in Curved Spacetime

-

Toroidal Compactification

Superconductivity

-

-

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

Topology

-

Units, Constants and Useful Formulas

-

-

Last modified: February 7, 2019


Index Notation (Tensors and Matrices)
-------------------------------------

Problems With Matrices
----------------------

1.  They can be big and difficult to manipulate.

2.  They do not commute so the order of multiplication 
    is important.

3.  Some things cannot be represented by matrices or 
    matrix algebra.

Consider the 2D rotation matrix:

 -  -      -   -     -          - -   -   
| x¹ | -> | x^1' | = |  cosθ sinθ || x¹ |
| x² |    | x^2' |   | -sinθ cosθ || x² |
 -  -      -   -     -          - -   -  

                     -                -
                  = |  cosθx¹ + sinθx² |
                    | -sinθx¹ + cosθx² |
                     -                -

Rewrite the rotation matrix as:

 -         -
| Λ^1'₁  Λ^1'₂ |
| Λ^2'₁  Λ^2'₂ |
 -         -

xⁱ -> x^i' = ΣΛ^i'_jx^j  
          ^j

       = Λ^i'_jx^j using Einstein's notation.

       = Λ^i'₁x¹ + Λ^i'₂x²

So we get:

x^1'  = cosθx¹ + sinθx²

and,

x^2'  = -sinθx¹ + cosθx² 

In index notation the order doesn't matter.  Thus,

x^jΛ^i'_j ≡ Λ^i'_jx^j

However, this not true for matrices.

 -   -     -  -  -          -
| x^1' | = | x¹ ||  cosθ sinθ |
| x^2' |   | x² || -sinθ cosθ |
 -   -     -  -  -          -

which is an illegal operation.

Converting Index to Matrix Notation
-----------------------------------

To get the matrix form we can rearrange the terms
but we need to ensure that like indeces are adjacent 
to each other before contraction.  This is because
the product of 2 matrices is only defined if the
number of columns in the first matrix is equal to
the number of rows in the second matrix, i.e.

         (m x n)(n x m) = m x k

Therefore,  A^m_nBⁿ_m will correspond to a legal matrix 
operation while Bⁿ_mA^m_n will not.

Example:

M^μν = A^μ_σA^ν_ρM^σρ:

    = A^ν_ρA^μ_σM^σρ:

    = A^ν_ρB^μρ where B^μρ = A^μ_σM^σρ

    = A^ν_ρ(B^ρμ)^T 

    = (AB^T)^νμ

    = ((AB^T)^νμ)^T = (BA^T)^μν

But B = AM.  Therefore,

M' = AMA^T