Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate...

tensor index notation and the closely related but distinct basis-independent abstract index notation. An index that is summed over is a summation index, in...

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

Wikimedia Commons has media related to Penrose graphical notation. Abstract index notation Angular momentum diagrams (quantum mechanics) Braided monoidal...

basis to reflect important physical aspects of the spacetime. The abstract index notation denotes tensors as if they were represented by their coefficients...

of basis elements, and requires no symbols for the indices. The abstract index notation is a way to write tensors such that the indices are no longer thought...

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with...

history of the abstract theory see also multilinear algebra. Ricci calculus The earliest foundation of tensor theory – tensor index notation. Order of a...

{\displaystyle g_{\mu \nu }} themselves as the metric (see, however, abstract index notation). With the quantities d x μ {\displaystyle dx^{\mu }} being regarded...

general contraction operation is sometimes called the trace. In tensor index notation, the basic contraction of a vector and a dual vector is denoted by f...

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...

measures the noncommutativity of the second covariant derivative. In abstract index notation, R d c a b Z c = ∇ a ∇ b Z d − ∇ b ∇ a Z d . {\displaystyle...

Note: General relativity articles using tensors will use the abstract index notation. The principle of general covariance was one of the central principles...

or no columns, called an empty matrix. The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written...

component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties...

of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations). Charts in an atlas...

for example we cannot lower the index of a ( 0 , 2 ) {\displaystyle (0,2)} tensor.) Ricci calculus Einstein notation Metric tensor Musical isomorphism...

Antisymmetric permutation object acting on tensors Ricci calculus – Tensor index notation for tensor-based calculations Symmetric tensor – Tensor invariant under...

tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket notation Multilinear subspace learning Multivector Geometric algebra Clifford algebra...

the basis v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} . In abstract index notation, R i c a b = R c b c a = R c a c b . {\displaystyle \mathrm {Ric}...

displaying wikidata descriptions as a fallback Ricci calculus – Tensor index notation for tensor-based calculations Spinor field – Geometric structurePages...

\nu }} is the energy–momentum tensor. All tensors are written in abstract index notation. Matching the theory's prediction to observational results for...

i = j ] . {\displaystyle \delta _{ij}=[i=j].} Often, a single-argument notation δ i {\displaystyle \delta _{i}} is used, which is equivalent to setting...

{x}^{i}}}} This is the explicit form of the covariant transformation rule. The notation of a normal derivative with respect to the coordinates sometimes uses a...

}F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0} or using the index notation with square brackets[note 1] for the antisymmetric part of the tensor:...

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:...

\{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...

a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as G = R − 1 2 g R , {\displaystyle \mathbf {G} =\mathbf...

ones mixed. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. A given contravariant index of a tensor...

order topology on the unit square Lexicographic ordering in tensor abstract index notation Lexicographically minimal string rotation Leximin order Long line...