Articles containing proofs | Tensors | Linear algebra | Permutations
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations. The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis: where each index i1, i2, ..., in takes values 1, 2, ..., n. There are nn indexed values of εi1i2...in, which can be arranged into an n-dimensional array. The key defining property of the symbol is total antisymmetry in the indices. When any two indices are interchanged, equal or not, the symbol is negated: If any two indices are equal, the symbol is zero. When all indices are unequal, we have: where p (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble i1, i2, ..., in into the order 1, 2, ..., n, and the factor (−1)p is called the sign or signature of the permutation. The value ε1 2 ... n must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose ε1 2 ... n = +1, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article. The term "n-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol n matches the dimensionality of the vector space in question, which may be Euclidean or non-Euclidean, for example, or Minkowski space. The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems; however it can be interpreted as a tensor density. The Levi-Civita symbol allows the determinant of a square matrix, and the cross product of two vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation. (Wikipedia).
Kronecker delta and Levi-Civita symbol | Lecture 7 | Vector Calculus for Engineers
Definition of the Kronecker delta and the Levi-Civita symbol (sometimes called the permutation symbol or Levi-Civita tensor). The relationship between the Kronecker delta and the Levi-Civita symbol is discussed. Join me on Coursera: https://www.coursera.org/learn/vector-calculus-engin
From playlist Vector Calculus for Engineers
Levi-Civita and Kronecker: A Remarkable Relationship | Deep Dive Maths
There is a remarkable relationship between the product of two Levi-Civita symbols and the determinant of a matrix with the Kronecker delta as elements. After defining the Levi-Civita symbol and the Kronecker delta, I show how to derive this relationship using permutation matrices and the
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Cross Products Using Levi Civita Symbol
Everyone has their favorite method of calculating cross products. Today I go over the way I was taught, and then a more formal way of doing cross products by using the levi civita tensor.
From playlist Math/Derivation Videos
Tensor Calculus for Physics Ep. 14 | Covariant Curl
Today we derive the expression for curl in a general covariant notation. We do this by promoting vectors to covariant vectors, derivatives to covariant derivatives, and also promote the levi-civita symbol to an actual tensor. This series is based off "Tensor Calculus for Physics" by Dwigh
From playlist New To Tensors? Start Here
Tensor Calculus Lecture 7c: The Levi-Civita Tensors
This course will eventually continue on Patreon at http://bit.ly/PavelPatreon Textbook: http://bit.ly/ITCYTNew Errata: http://bit.ly/ITAErrata McConnell's classic: http://bit.ly/MCTensors Table of Contents of http://bit.ly/ITCYTNew Rules of the Game Coordinate Systems and the Role of Te
From playlist Introduction to Tensor Calculus
Since we just learn the pronoun ne, let's learn another one: ci. This has a lot of uses, as ci locativo, ci argomentale, and more. Let's learn how to use it! Script by Patrizia Farina, Professor of Italian at Western Connecticut State University and Purchase College. Watch the whole Ital
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Divergence of the cross product of two vectors (proof) | Lecture 22 | Vector Calculus for Engineers
An example of how to prove a vector calculus identity using the Levi-Civita symbol and the Kronecker delta. Join me on Coursera: https://www.coursera.org/learn/vector-calculus-engineers Lecture notes at http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf Subscribe to my ch
From playlist Vector Calculus for Engineers
Cassini's identity | Lecture 7 | Fibonacci Numbers and the Golden Ratio
Derivation of Cassini's identity, which is a relationship between separated Fibonacci numbers. The identity is derived using the Fibonacci Q-matrix and determinants. Join me on Coursera: https://www.coursera.org/learn/fibonacci Lecture notes at http://www.math.ust.hk/~machas/fibonacci.pd
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Vector Triple Product | Lecture 10 | Vector Calculus for Engineers
The vector triple product identity is proved using the Levi-Civita symbol and the Einstein summation convention. Join me on Coursera: https://www.coursera.org/learn/vector-calculus-engineers Lecture notes at http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf Subscribe to
From playlist Vector Calculus for Engineers
👉 Learn how to multiply polynomials. To multiply polynomials, we use the distributive property. The distributive property is essential for multiplying polynomials. The distributive property is the use of each term of one of the polynomials to multiply all the terms of the other polynomial.
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Vector Identities | Lecture 8 | Vector Calculus for Engineers
Four vector identities are presented: (1) Scalar triple product; (2) Vector triple product; (3) Scalar quadruple product; (4) Vector quadruple product. The tools required to prove them are discussed. Join me on Coursera: https://www.coursera.org/learn/vector-calculus-engineers Lecture n
From playlist Vector Calculus for Engineers
Can you prove the Why equation? Part 2
This amazingly simple equation can be proved in just two short lines. Watch the video to learn how.
From playlist Summer of Math Exposition Youtube Videos
Scalar Triple Product | Lecture 9 | Vector Calculus for Engineers
The scalar triple product identity is proved using the Levi-Civita symbol and the Einstein summation convention. Join me on Coursera: https://www.coursera.org/learn/vector-calculus-engineers Lecture notes at http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf Subscribe to
From playlist Vector Calculus for Engineers
http://bit.ly/PavelPatreon Textbook: http://bit.ly/ITCYTNew Errata: http://bit.ly/ITAErrata McConnell's classic: http://bit.ly/MCTensors Table of Contents of http://bit.ly/ITCYTNew Rules of the Game Coordinate Systems and the Role of Tensor Calculus Change of Coordinates The Tensor Desc
From playlist Introduction to Tensor Calculus
Introduction to Polygons - Geometry
Learn the definition of polygon - a very important shape in geometry. When a polygon has a small number of sides, there is a word you use instead of "polygon". We teach you the names of polygons with 3 to 10 sides. Geometer: Louise McCartney Artwork: Kelly Vivanco Written by Michael Ha
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