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In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity.
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions over a manifold, usually more specifically as functions of the coordinates on the manifold. This allows intuitive manipulation of expressions with familiarity of only a limited set of rules.
A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
The following can be shown independent of the distance between roots of the indicial polynomial: Let be a -fold root of the indicial polynomial relative to . Then the part of the fundamental system corresponding to α {\displaystyle \alpha } is given by the μ {\displaystyle \mu } linearly independent formal solutions
The notation and terminology are relatively obsolete today. Its uses in physics include continuum mechanics and electromagnetism. In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over- or underbars.
Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.
Abstract index notation (also referred to as slot-naming index notation) [1] is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. [2] The indices are mere placeholders, not related to any basis and, in particular, are non-numerical.