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Einstein notation, the Glossary

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.^[1]

59 relations: Abstract index notation, Addition, Albert Einstein, Basis (linear algebra), Bra–ket notation, Change of basis, Coefficient, Coordinate vector, Cotangent space, Covariance and contravariance of vectors, Cross product, Degenerate bilinear form, DeWitt notation, Differential geometry, Dot product, Dual basis, Dual space, Euclidean distance, Exponentiation, Free variables and bound variables, General relativity, Greek alphabet, Indexed family, Infinite set, Inner product space, Invariant (mathematics), Invertible matrix, Isomorphism, Kronecker delta, Latin alphabet, Levi-Civita symbol, Linear algebra, Linear form, Linear map, Lorentz scalar, Lorentz transformation, Mathematical physics, Mathematics, Matrix (mathematics), Matrix multiplication, Metric tensor, Minkowski space, Orthogonal basis, Orthonormal basis, Outer product, PDF, Penrose graphical notation, Raising and lowering indices, Ricci calculus, Riemannian manifold, ... Expand index (9 more) »
Multilinear algebra

Abstract index notation

Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. Einstein notation and Abstract index notation are mathematical notation and tensors.

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Addition

Addition (usually signified by the plus symbol) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. Einstein notation and Addition are mathematical notation.

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Albert Einstein

Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who is widely held as one of the most influential scientists. Best known for developing the theory of relativity, Einstein also made important contributions to quantum mechanics. His mass–energy equivalence formula, which arises from relativity theory, has been called "the world's most famous equation".

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Basis (linear algebra)

In mathematics, a set of vectors in a vector space is called a basis (bases) if every element of may be written in a unique way as a finite linear combination of elements of.

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Bra–ket notation

Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. Einstein notation and Bra–ket notation are mathematical notation.

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Change of basis

In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates.

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Coefficient

In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or an expression.

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Coordinate vector

In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis.

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Cotangent space

In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Einstein notation and cotangent space are tensors.

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Covariance and contravariance of vectors

In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. Einstein notation and covariance and contravariance of vectors are Riemannian geometry and tensors.

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Cross product

In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times.

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Degenerate bilinear form

In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V&hairsp) given by is not an isomorphism.

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DeWitt notation

Physics often deals with classical models where the dynamical variables are a collection of functions α over a d-dimensional space/spacetime manifold M where α is the "flavor" index. Einstein notation and DeWitt notation are mathematical notation.

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Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. Einstein notation and Differential geometry are mathematical physics.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product means literally "product with a scalar as a result". Einstein notation and dot product are tensors.

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Dual basis

In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimension of V), the dual set of B is a set B^* of vectors in the dual space V^* with the same index set I such that B and B^* form a biorthogonal system.

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Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

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Euclidean distance

In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them.

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Exponentiation

In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power.

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Free variables and bound variables

In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Einstein notation and free variables and bound variables are mathematical notation.

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General relativity

General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. Einstein notation and general relativity are Albert Einstein.

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Greek alphabet

The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC.

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Indexed family

In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. Einstein notation and indexed family are mathematical notation.

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Infinite set

In set theory, an infinite set is a set that is not a finite set.

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Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product.

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Invariant (mathematics)

In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects.

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Invertible matrix

In linear algebra, an -by- square matrix is called invertible (also nonsingular, nondegenerate or rarely regular) if there exists an -by- square matrix such that\mathbf.

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Isomorphism

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.

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Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. Einstein notation and Kronecker delta are mathematical notation.

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Latin alphabet

The Latin alphabet, also known as the Roman alphabet, is the collection of letters originally used by the ancient Romans to write the Latin language.

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Levi-Civita symbol

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers, for some positive integer. Einstein notation and Levi-Civita symbol are tensors.

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Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices.

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Linear form

In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of scalars (often, the real numbers or the complex numbers).

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Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication.

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Lorentz scalar

In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation.

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Lorentz transformation

In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. Einstein notation and Lorentz transformation are mathematical physics.

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Mathematical physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Matrix (mathematics)

In mathematics, a matrix (matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.

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Matrix multiplication

In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices.

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Metric tensor

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. Einstein notation and metric tensor are Riemannian geometry and tensors.

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Minkowski space

In physics, Minkowski space (or Minkowski spacetime) is the main mathematical description of spacetime in the absence of gravitation.

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Orthogonal basis

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal.

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Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

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Outer product

In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector.

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PDF

Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems.

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Penrose graphical notation

In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. Einstein notation and Penrose graphical notation are mathematical notation and tensors.

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Raising and lowering indices

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Einstein notation and raising and lowering indices are tensors.

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Ricci calculus

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. Einstein notation and Ricci calculus are tensors.

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Riemannian manifold

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Einstein notation and Riemannian manifold are Riemannian geometry.

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Scalar (physics)

Scalars (or scalar quantities) are physical quantities that are unaffected by changes to a vector space basis (i.e., a coordinate system transformation).

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Set (mathematics)

In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

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Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns.

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Summation

In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. Einstein notation and summation are mathematical notation.

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Tangent space

In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions.

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Tensor

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Einstein notation and tensor are tensors.

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Tensor product

In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted.

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Trace (linear algebra)

In linear algebra, the trace of a square matrix, denoted, is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of.

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Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.

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References

[1] https://en.wikipedia.org/wiki/Einstein_notation

Also known as Einstein Summation convention, Einstein convention, Einstein rule, Einstein summation, Einstein summation notation, Einstein's summation convention, Summation convention.

, Scalar (physics), Set (mathematics), Square matrix, Summation, Tangent space, Tensor, Tensor product, Trace (linear algebra), Vector space.

Einstein notation, the Glossary

Table of Contents

See also

Multilinear algebra

References