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Manner of referring to elements of arrays or tensors
In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies
Index_notation
Mathematical notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory
Multi-index_notation
Mathematical notation for tensors and spinors
Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate
Abstract_index_notation
Tensor index notation for tensor-based calculations
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with
Ricci_calculus
Concise notation for large or small numbers
referred to as scientific form or standard index form, or standard form in the United Kingdom. This base ten notation is commonly used by scientists, mathematicians
Scientific_notation
Shorthand notation for tensor operations
implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however
Einstein_notation
Notation system for crystal lattice planes
Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of
Miller_index
Association of one output to each input
value of the function f at the point (x0, t0). Index notation may be used instead of functional notation. That is, instead of writing f (x), one writes
Function_(mathematics)
Graphical notation for multilinear algebra calculations
In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions
Penrose_graphical_notation
Atoms of the same element, but different mass
element symbol is used, e.g. "C" for carbon, it is standard to use "AZE notation" which has the form A ZE, where A is the mass number written as a superscript
Isotope
Operation in mathematics
2x2; often 3x3 or 4x4 are used, but any size is allowed. In simple index notation, this is written ∑ j = 1 2 a i j × b j k = c i k {\textstyle \sum
Tensor_contraction
Mathematical Concept
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
Voigt_notation
Exterior algebraic map taking tensors from p forms to n-p forms
}(dy\wedge dz)&=dt\wedge dx\,.\end{aligned}}} These are summarized in the index notation as ⋆ ( d x μ ) = η μ λ ε λ ν ρ σ 1 3 ! d x ν ∧ d x ρ ∧ d x σ , ⋆ ( d
Hodge_star_operator
Algebraic object with geometric applications
abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates. This notation captures
Tensor
Antisymmetric permutation object acting on tensors
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:
Levi-Civita_symbol
Array of numbers
or no columns, called an empty matrix. The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written
Matrix_(mathematics)
Tensor that describes the 4D geometry of spacetime
g_{\rho \sigma }.} The metric tensor plays a key role in index manipulation. In index notation, the coefficients g μ ν {\displaystyle g_{\mu \nu }} of
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Polynomial with only one term
substituting by 1 the extra variable. The multi-index notation is often useful for having a compact notation, specially when there are more than two or three
Monomial
Isomorphism between the tangent and cotangent bundles of a manifold
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
Musical_isomorphism
Algebra associated to any vector space
V ) {\displaystyle t\in A^{r}(V)\subset T^{r}(V)} can be written in index notation with the Einstein summation convention as t = t i 1 i 2 ⋯ i r e i 1
Exterior_algebra
Mathematical operation on vectors in 3D space
mathematics, the wedge notation a ∧ b is often used (in conjunction with the name vector product), although in pure mathematics such notation is usually reserved
Cross_product
Tensor in differential geometry
v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} . In abstract index notation, Ric a b = R c b c a = R c a c b . {\displaystyle \operatorname
Ricci_curvature
Expression that may be integrated over a region
dependent is zero. A common notation for the wedge product of elementary k {\displaystyle k} -forms is so called multi-index notation: in an n {\displaystyle
Differential_form
Tensor equal to the negative of any of its transpositions
Antisymmetric permutation object acting on tensors Ricci calculus – Tensor index notation for tensor-based calculations Symmetric tensor – Tensor invariant under
Antisymmetric_tensor
Tensor describing energy momentum density in spacetime
superscripted variables (not exponents; see Tensor index notation and Einstein summation notation). The four coordinates of an event of spacetime x are
Stress–energy_tensor
Algebraic operation on coordinate vectors
specified with respect to an orthonormal basis, is defined, in summation notation, as: a ⋅ b = ∑ i = 1 n a i b i = a 1 b 1 + a 2 b 2 + ⋯ + a n b n {\displaystyle
Dot_product
Matrix operation which flips a matrix over its diagonal
another matrix, called the transpose of A and often denoted AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician
Transpose
Specification of a derivative along a tangent vector of a manifold
coordinate-free language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension
Covariant_derivative
Convention where symbols represent concepts
Bra–ket notation, or Dirac notation, is an alternative representation of probability distributions in quantum mechanics. Tensor index notation is used
Notation_system
Notation used for Weyl spinors
indices, i.e. "index free notation", an overbar is retained on right-handed spinor, since ambiguity arises between chirality when no index is indicated
Van_der_Waerden_notation
of tensor theory – tensor index notation. Order of a tensor The components of a tensor with respect to a basis is an indexed array. The order of a tensor
Glossary_of_tensor_theory
Tensor field in Riemannian geometry
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 R^{d}{}_{cab}Z^{c}=\nabla
Riemann_curvature_tensor
Second order tensor in vector algebra
algebra, a dyadic or dyadic tensor is a second-order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two
Dyadics
Branch of mathematics
tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket notation Multilinear subspace learning Multivector Geometric algebra Clifford algebra
Multilinear_algebra
Mathematical object that describes the electromagnetic field in spacetime
}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:
Electromagnetic_tensor
Assignment of a tensor continuously varying across a region of space
bundle – Construction in differential topology Ricci calculus – Tensor index notation for tensor-based calculations Spinor field – Geometric structurePages
Tensor_field
Continuous surjection satisfying a local triviality condition
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Fiber_bundle
Method for specifying point positions
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Coordinate_system
Musical symbols are marks and symbols in musical notation that indicate various aspects of how a piece of music is to be performed. There are symbols to
List_of_musical_symbols
Set of vectors used to define coordinates
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 = [
Basis_(linear_algebra)
Tensor having both covariant and contravariant indices
ones mixed. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. A given contravariant index of a tensor
Mixed_tensor
Property of a mathematical space
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Dimension
Mathematical function, in linear algebra
Victor (2001) [1994], "Index theory", Encyclopedia of Mathematics, EMS Press: "The main question in index theory is to provide index formulas for classes
Linear_map
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
i = j ] . {\displaystyle \delta _{ij}=[i=j].} Often, a single-argument notation δ i {\displaystyle \delta _{i}} is used, which is equivalent to setting
Kronecker_delta
Covariant derivative of the metric tensor
, Y , Z {\displaystyle X,Y,Z} arbitrary vector fields. In abstract index notation, this reads Q a b c = ∇ a g b c {\displaystyle Q_{abc}=\nabla _{a}g_{bc}}
Nonmetricity_tensor
Mathematical operation in linear algebra
matrices are italic (they are numbers from a field), e.g. A and a. Index notation is often the clearest way to express definitions, and is used as standard
Matrix_multiplication
Notation of differential calculus
differential calculus, there is no single standard notation for differentiation. Instead, several notations for the derivative of a function or a dependent
Notation_for_differentiation
Vector behavior under coordinate changes
covectors) are said to be contravariant. In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Theory of gravitation as curved spacetime
}} is the stress–energy tensor. All tensors are written in abstract index notation. Matching the theory's prediction to observational results for planetary
General_relativity
Topics referred to by the same term
of a string of characters Suffix notation, a notation for manipulating vector quantities, also known as index notation Suffix array, an array of integers
Suffix_(disambiguation)
Operation on differential forms
generalized for any pseudo-Riemannian manifold, and written in coordinate-free notation as follows: grad f ≡ ∇ f = ( d f ) ♯ div F ≡ ∇ ⋅ F = ⋆ d ⋆ ( F ♭ )
Exterior_derivative
Differential form of degree one or section of a cotangent bundle
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
One-form
Conserved physical quantity; rotational analogue of linear momentum
_{z}\wedge \mathbf {e} _{x}\,,\end{aligned}}} or more compactly in index notation: L i j = x i p j − x j p i . {\displaystyle L_{ij}=x_{i}p_{j}-x_{j}p_{i}\
Angular_momentum
Representation of a tensor in Euclidean space
ayey (a vector), and similarly for x and z. A more general notation is tensor index notation, which has the flexibility of numerical values rather than
Cartesian_tensor
Origin and evolution of the symbols used to write equations and formulas
mathematical notation covers the introduction, development, and cultural diffusion of mathematical symbols and the conflicts between notational methods that
History of mathematical notation
History_of_mathematical_notation
Abbreviation in the fields of special and general relativity
four-dimensional spacetime. General four-tensors are usually written in tensor index notation as A ν 1 , ν 2 , . . . , ν m μ 1 , μ 2 , . . . , μ n {\displaystyle
Four-tensor
Mathematical operation on vector spaces
differentiable, then a */ b is differentiable. However, these kinds of notation are not universally present in array languages. Other array languages may
Tensor_product
Straight path on a curved surface or a Riemannian manifold
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Geodesic
Approach to general relativity
to reflect important physical aspects of the spacetime. The abstract index notation denotes tensors as if they were represented by their coefficients with
Tetrad_formalism
Stress case in finite deformations
{F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}~.} In index notation with respect to an orthonormal basis, S I L = J F I k − 1 F L m
Piola–Kirchhoff stress tensors
Piola–Kirchhoff_stress_tensors
Topological space that locally resembles Euclidean space
such as hearing the shape of a drum and some proofs of the Atiyah–Singer index theorem. Infinite dimensional manifolds The definition of a manifold can
Manifold
Coordinate-free definition of a tensor
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Tensor_(intrinsic_definition)
Note: General relativity articles using tensors will use the abstract index notation. The principle of general covariance was one of the central principles
Mathematics of general relativity
Mathematics_of_general_relativity
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
\end{aligned}}} It is common in rigid body mechanics to use notation that explicitly identifies the x {\displaystyle x} , y {\displaystyle y}
Moment_of_inertia
Equations describing elastic deformation
forces that could be expressed as potentials) on the boundary are (using index notation) the equilibrium equation: σ i j , i = 0 {\displaystyle \sigma _{ij
Stress_functions
Tensor used in general relativity
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 {\boldsymbol {G}}={\boldsymbol
Einstein_tensor
Measure of the curvature of a pseudo-Riemannian manifold
v_{3}\right)k\left(v_{1},v_{4}\right)\end{aligned}}} In tensor component notation, this can be written as C i k ℓ m = R i k ℓ m + 1 n − 2 ( R i m g k ℓ −
Weyl_tensor
Canonical connection on a pseudo-Riemannian manifold
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Levi-Civita_connection
Array of numbers describing a metric connection
the same notation as tensors with index notation, they do not transform like tensors under a change of coordinates. Contracting the upper index with either
Christoffel_symbols
Structure defining distance on a manifold
is increased by du units, and v is increased by dv units. Using matrix notation, the first fundamental form becomes d s 2 = [ d u d v ] [ E F F G ] [ d
Metric_tensor
Force needed to pull a spring grows linearly with distance
sum of a constant tensor and a traceless symmetric tensor. Thus in index notation: ε i j = ( 1 3 ε k k δ i j ) + ( ε i j − 1 3 ε k k δ i j ) {\displaystyle
Hooke's_law
Construct in differenital geometry
dx^{i}.} The point of the notation is to distinguish the indices j, k, which run over the n dimensions of the fiber, from the index i, which runs over the
Metric_connection
Mapping from p forms to p-1 forms
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Interior_product
Relativistic vector field
field, depending upon the choice of gauge. This article uses tensor index notation and the Minkowski metric sign convention (+ − − −). See also covariance
Electromagnetic four-potential
Electromagnetic_four-potential
Equations of motion for viscous fluids
{u} )]+\rho \mathbf {a} .} in index notation, the equation can be written as Navier–Stokes momentum equation (index notation) ρ ( ∂ u i ∂ t + u k ∂ u i ∂
Navier–Stokes_equations
Derivative used in gauge theories
after choosing a frame for the fields involved, often in the form of index notation. There are many ways to understand the gauge covariant derivative. The
Gauge_covariant_derivative
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Symmetrization
Function that is invariant under all permutations of its variables
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Symmetric_function
Branch of mathematics
popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation g {\displaystyle g} for a Riemannian metric, and Γ {\displaystyle \Gamma
Differential_geometry
Spacetime modeled by four pointwise-orthonormal vector fields
in 1928 and by Hermann Weyl in 1929. The index notation for tetrads is explained in tetrad (index notation). Frame fields of a Lorentzian manifold always
Frame fields in general relativity
Frame_fields_in_general_relativity
Universal construction in multilinear algebra
was actually one and the same thing as ∇ {\displaystyle \nabla } ; and notational sloppiness here would lead to utter chaos. To strengthen this: the tensor
Tensor_algebra
Material property in strain-stress relationship
function. Hooke's law may be written in terms of tensor components using index notation as σ i j = 2 μ ε i j + λ δ i j ε k k , {\displaystyle \sigma _{ij}=2\mu
Lamé_parameters
System of moving vectors in differential geometry
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Parallel_transport
Representation of mechanical stress at every point within a deformed 3D object
_{1}+\sigma _{32}\mathbf {e} _{2}+\sigma _{33}\mathbf {e} _{3},} In index notation this is T ( e i ) = T j ( e i ) e j = σ i j e j . {\displaystyle \mathbf
Cauchy_stress_tensor
Generalized alphabetical order
topology on the unit square Lexicographic ordering in tensor abstract index notation Lexicographically minimal string rotation Leximin order Long line (topology)
Lexicographic_order
Visual representation of music
Musical notation is any system used to visually represent music. Systems of notation generally represent the elements of a piece of music that are considered
Musical_notation
Data type that represents an ordered collection of elements (values or variables)
use to define such types and declare array variables, and special notation for indexing array elements. For example, in the Pascal programming language
Array_(data_type)
Type of physical quantity
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Pseudotensor
Non-tensorial representation of the spin group
form on a complex vector space is equivalent to the standard one, this notation is often used whenever dimℂ(V) = n. If n = 2k is even, then Cℓn(ℂ) is isomorphic
Spinor
Basis used to express spherical tensors
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Spherical_basis
General relativity equation
Throughout, this article uses the natural units c = G = 1, and tensor index notation. The Mathisson–Papapetrou–Dixon (MPD) equations for a mass m {\displaystyle
Mathisson–Papapetrou–Dixon equations
Mathisson–Papapetrou–Dixon_equations
Tensor invariant under permutations of vectors it acts on
the operator is omitted: T1T2 = T1 ⊙ T2. In some cases an exponential notation is used: v ⊙ k = v ⊙ v ⊙ ⋯ ⊙ v ⏟ k times = v ⊗ v ⊗ ⋯ ⊗ v ⏟ k times =
Symmetric_tensor
Action of a massive abelian gauge field
W bosons. This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors. The field involved is a complex 4-potential
Proca_action
Electromagnetic stress
Maxwell) is the stress tensor of an electromagnetic field. In tensor index notation it is given by: σ i j = ε 0 E i E j + 1 μ 0 B i B j − ( ε 0 2 E 2 +
Maxwell_stress_tensor
Specialized notation for multivariable calculus
notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. Two competing notational conventions
Matrix_calculus
Method of notation of very large integers
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. In his 1947 paper, R. L
Knuth's_up-arrow_notation
Branch of physics which studies the behavior of materials modeled as continuous media
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Continuum_mechanics
Theory of interwoven space and time by Albert Einstein
disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since n {\displaystyle n} depends on wavelength
Special_relativity
Ways of writing certain laws of physics
equations, one for each value of β. Using the antisymmetric tensor notation and comma notation for the partial derivative (see Ricci calculus), the second equation
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Millennium Prize Problem
(see smooth function) such that, for every multi-index α {\displaystyle \alpha } (see multi-index notation) and any K > 0 {\displaystyle K>0} , there exists
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
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INDEX NOTATION
INDEX NOTATION
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INDEX NOTATION
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