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...

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...

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...

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

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

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...

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...

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...

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

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

Einstein summation notation: any index may appear at most twice and furthermore a raised index must contract with a lowered index. With these rules we...

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

such as causal structure, and quantum effects. The book uses the abstract index notation for tensors. It treats spinors, the variational-principle formulation...

(with Brian W. Aldiss) (1999) Concepts Twistor theory Spin network Abstract index notation Black hole bomb Geometry of spacetime Cosmic censorship Weyl curvature...

mathematical notation covers the introduction, development, and cultural diffusion of mathematical symbols and the conflicts between notational methods that...

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

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

\{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...

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

contributions  Moore–Penrose inverse Twistor theory Spin network Abstract index notation Black hole bomb Geometry of spacetime Cosmic censorship Illumination...

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...

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...

manifold M, and xa are coordinates in the fiber Mp. Using the abstract index notation, let a, b, c,… refer to Mp and μ, ν,… refer to the tangent bundle...

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}...

Scope Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)...

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...

elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows...