differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies...

Summation notation may refer to: Capital-sigma notation, mathematical symbol for summation Einstein notation, summation over like-subscripted indices This...

with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention to compensate...

contrast, a dyad is specifically a dyadic tensor of rank one. Einstein notation This notation is based on the understanding that whenever a multidimensional...

For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal...

set equal to each other and summed over. In Einstein notation this summation is built into the notation. The result is another tensor with order reduced...

vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation. The Levi-Civita symbol is most often used in three and four dimensions...

(as opposed to those of covectors) are said to be contravariant. In Einstein notation (implicit summation over repeated index), contravariant components...

The Einstein tensor G {\displaystyle {\boldsymbol {G}}} is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is...

forms Component-free treatment of tensors Cramer's rule Dual space Einstein notation Exterior algebra Inner product Outer product Kronecker delta Levi-Civita...

In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that ( ∑ i = 1 n a i c i ) ( ∑ j = 1...

in the article del in cylindrical and spherical coordinates. Using Einstein notation we can consider the divergence in general coordinates, which we write...

which will also be mixed. Covariance and contravariance of vectors Einstein notation Ricci calculus Tensor (intrinsic definition) Two-point tensor D.C...

physicist Albert Einstein's formula E = m c 2 {\displaystyle E=mc^{2}} is the quantitative representation in mathematical notation of mass–energy equivalence...

also become smaller: 1 Kelvin per m becomes 0.001 Kelvin per mm. In Einstein notation, contravariant vectors and components of tensors are shown with superscripts...

matrix A by producing another matrix, often denoted by AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician...

associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas...

constant G {\displaystyle G} will be kept explicit. This article employs the Einstein summation convention, where repeated indices are automatically summed over...

\delta _{ab\dots }^{cd\dots }} is the generalized Kronecker delta, and the Einstein summation convention is in use. More generally, irrespective of the number...

\{k\in \mathbb {N} :1\leq k\leq n\}} , and u i {\displaystyle u_{i}} is a notation for the image of i {\displaystyle i} by the function/vector u {\displaystyle...

the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article. The components...

. Note that the μ and ν summation indices range from 0 to 3: see Einstein notation. (Some authors alternatively use the negative metric signature of...

Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory...

{\displaystyle \partial _{i}f} and f i {\displaystyle f_{i}}  : Written with Einstein notation, where repeated indices (i) are summed over. The gradient (or gradient...

the basis as v = v i e i {\displaystyle v=v^{i}e_{i}} using Einstein summation notation, i.e., v {\displaystyle v} has components v i {\displaystyle...

j}y_{j},} for i = 1, ..., n. This formula may be concisely written in matrix notation. Let A be the matrix of the a i , j {\displaystyle a_{i,j}} , and X = [...

solutions of the equation. Although succinct when written out using Einstein notation, hidden within the Ricci tensor and Ricci scalar are exceptionally...

tensor theory Scope Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad...

This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign...

differentiable, then a */ b is differentiable. However, these kinds of notation are not universally present in array languages. Other array languages may...