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In a light-cone coordinate system, two of the coordinates are null vectors and all the other coordinates are spatial. The former can be denoted + and and the latter .. Assume we are working with a (d,1) Lorentzian signature.
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.
Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
This section uses Einstein notation, including Einstein summation convention. See also Ricci calculus for a summary of tensor index notations, and raising and lowering indices for definition of superscript and subscript indices, and how to switch between them. The Minkowski metric tensor η here has metric signature (+ − − −).
In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices as in = A covector or cotangent vector has components that co-vary with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix ...
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In orthogonal coordinates, the elastic energy per unit volume due to strain is thus a sum of contributions: =, where is a 4th rank tensor, called the elastic tensor or stiffness tensor [3] which is a generalization of the elastic moduli of mechanical systems, and is the strain tensor (Einstein summation notation has been used to imply summation ...
For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions ...