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Encyclopedia > Einstein notation
For other topics related to Einstein, see Einstein (disambiguation).

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. It was introduced by Albert Einstein in 1916 [1]. A number of topics are named Einstein; most are related to the noted physicist Albert Einstein. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ...


According to this convention, when an index variable appears twice in a single term, once in an upper (superscript) and once in a lower (subscript) position, it implies that we are summing over all of its possible values. In typical applications, the indices are 1,2,3 (representing the three dimensions of physical Euclidean space), or 0,1,2,3 or 1,2,3,4 (representing the four dimensions of space-time, or Minkowski space), but they can have any range, even (in some applications) an infinite set. Abstract index notation is an improvement of Einstein notation. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In set theory, an infinite set is a set that is not a finite set. ... Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...


In general relativity, the Greek alphabet and the Roman alphabet are used to distinguish whether summing over 1,2,3 or 0,1,2,3 (usually Roman, i, j, ... for 1,2,3 and Greek, μ, ν, ... for 0,1,2,3). As in sign conventions, the convention used in practice varies: Roman and Greek may be reversed. General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BCE. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel and consonant alike. ... The Latin alphabet, also called the Roman alphabet, is the most widely used alphabetic writing system in the world today. ...


Sometimes (as in general relativity), the index is required to appear once as a superscript and once as a subscript; in other applications, all indices are subscripts. See Dual vector space and Tensor product. General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...


It is important to keep in mind that no new physical laws or ideas result from using Einstein notation; rather, it merely helps in identifying relationships and symmetries often 'hidden' by more conventional notation.


In some fields, Einstein notation is referred to simply as index notation, or indicial notation. Additionally, the use of the implied summation of repeated indices is referred to as the Einstein Sum Convention. Index notation is used in mathematics to refer to the elements of matrices or the components of a vector. ... In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ...

Contents

Introduction

The basic idea of Einstein notation is very simple. It allows one to replace something bulky, such as:

y = c1x1 + c2x2 + c3x3 + ... + cnxn

typically written as:

y = sum_{i=1}^n c_ix_i

with something even simpler, in Einstein notation:

y = c_i x^i ,

In Einstein notation, indices such as i in the equation above can appear as either subscripts or superscripts. The position of the index has a specific meaning. It is important, of course, not to interpret an index appearing in the superscript position as if it were an exponent, which is the convention in standard algebra. Here, the superscripted i above the symbol x represents an integer-valued index running from 1 to n.


The virtue of Einstein notation is that an index appearing two or more times in a single term implies summation across that index, so that the summation symbol is unnecessary. Since the summation in effect "eliminates" the index over which the sum is taken, the summation index does not appear on the opposite side of the equals sign.


Vector representations

First, we can use Einstein notation in linear algebra to distinguish easily between row vectors and column vectors. We could, for example, use superscripted indices to represent the elements of column vectors, and subscripted indices to represent the elements of row vectors. Following this convention, then, Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...

mathbf{u} = u^i   mathrm{for}   i = 1, 2, 3, ... , M

represents an M × 1 column vector, and

mathbf{v} = v_j   mathrm{for}   j = 1, 2, 3, ... , N

represents a 1 × N row vector.


In mathematics and theoretical physics, and in particular general relativity, column vectors represent contravariant vectors whereas row vectors represent covariant vectors. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... In category theory, see covariant functor. ...


Matrix representation

Using standard notation, we can generate M × N matrix A by multiplying column vector u by row vector v: This article gives an overview of the various ways to perform matrix multiplication. ...

mathbf{A} = mathbf{u} cdot mathbf{v}

In Einstein notation, we have:

{A^i}_j = u^i cdot v_j = uv^i_j

Since i and j represent two different indices, and in this case over two different ranges M and N respectively, the indices are not eliminated by the multiplication. Both indices survive the multiplication to become the two indices of the newly-created matrix A.


Matrix multiplication

We can represent matrix multiplication as: This article gives an overview of the various ways to perform matrix multiplication. ...

C^i_k = A^i_j cdot B^j_k

This expression is equivalent to the more conventional (and less compact) notation:

mathbf{C} = mathbf{A} cdot mathbf{B} =sum_{j=1}^N A_{ij} B_{jk}

Vector dot product

In mechanics and engineering, vectors in 3D space are often described in relation to orthogonal unit vectors i, j and k. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

mathbf{u} = u_x mathbf{i} + u_y mathbf{j} + u_z mathbf{k}

If the basis vectors i, j, and k are instead expressed as e1, e2, and e3, a vector can be expressed in terms of a summation:

mathbf{u} = u_1 mathbf{e}_1 + u_2 mathbf{e}_2 + u_3 mathbf{e}_3 = sum_{i = 1}^3 u_i mathbf{e}_i

In Einstein notation, the summation symbol is omitted since the index i is repeated and we simply write

mathbf{u} = u_i mathbf{e}_i


Using e1, e2, and e3 instead of i, j, and k, together with Einstein notation, we obtain a concise algebraic presentation of vector and tensor equations. For example, In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

mathbf{u} cdot mathbf{v} = sum_{i = 1}^3 u_i mathbf{e}_i cdot sum_{j = 1}^3 v_j mathbf{e}_j = u_i mathbf{e}_i cdot v_j mathbf{e}_j

or equivalently:

mathbf{u} cdot mathbf{v} = sum_{i = 1}^3 sum_{j = 1}^3 u_i v_j ( mathbf{e}_i cdot mathbf{e}_j ) = u_i v_j ( mathbf{e}_i cdot mathbf{e}_j )

where

mathbf{e}_i cdot mathbf{e}_j = delta_{ij}

and  delta_{ij} is the Kronecker delta, which is equal to 1 when i = j, and 0 otherwise. It logically follows that this allows one j in the equation to be converted to an i, or one i to be converted to a j. Then, In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

mathbf{u} cdot mathbf{v} = u^i v^jdelta_{ij}= u^i v_i = u_j v^j

Vector cross product

For the cross product, For the crossed product in algebra and functional analysis, see crossed product. ...

mathbf{u} times mathbf{v}= sum_{j = 1}^3 u_j mathbf{e}_j times sum_{k = 1}^3 v_k mathbf{e}_k = u_j mathbf{e}_j times v_k mathbf{e}_k = u_j v_k (mathbf{e}_j times mathbf{e}_k ) = epsilon_{ijk} mathbf{e}_i u_j v_k

where mathbf{e}_j times mathbf{e}_k = epsilon_{ijk} mathbf{e}_i and  epsilon_{ijk} is the Levi-Civita symbol defined by: The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ...

epsilon_{ijk} = left{ begin{matrix} 0 & mbox{unless } i,j,k mbox{ are distinct} +1 & mbox{if } (i,j,k) mbox{ is an even permutation of } (1,2,3) -1 & mbox{if } (i,j,k) mbox{ is an odd permutation of } (1,2,3) end{matrix} right.

which recovers

mathbf{u} times mathbf{v} = (u_2 v_3 - u_3 v_2) mathbf{e}_1 + (u_3 v_1 - u_1 v_3) mathbf{e}_2 + (u_1 v_2 - u_2 v_1) mathbf{e}_3

from

mathbf{u} times mathbf{v}= epsilon_{ijk} mathbf{e}_i u_j v_k = sum_{i = 1}^3 sum_{j = 1}^3 sum_{k = 1}^3 epsilon_{ijk} mathbf{e}_i u_j v_k.

Additionally, if mathbf{w} = mathbf{u} times mathbf{v}, then mathbf{w} = epsilon_{ijk} mathbf{e}_i u_j v_k and  w_i = epsilon_{ijk} u_j v_k. This also highlights that when an index appears once on both sides of the equation, this implies a system of equations instead of a summation:

begin{matrix} w_1 = epsilon_{1jk} u_j v_k w_2 = epsilon_{2jk} u_j v_k w_3 = epsilon_{3jk} u_j v_k end{matrix}

Alternatively, this could be expressed as

mathbf{u} times mathbf{v}= mathbf{u} cdot epsilon cdot mathbf{v}

but, this isn't the notation Einstein used.


Abstract definitions

In the traditional usage, one has in mind a vector space V  with finite dimension n, and a specific basis of V. We can write the basis vectors as e1, e2, ..., en. Then if v is a vector in V, it has coordinates v1, ..., vn relative to this basis. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...


The basic rule is:

v = vi ei.

In this expression, it was assumed that the term on the right side was to be summed as i  goes from 1 to n, because the index i does not appear on both sides of the expression. (Or, using Einstein's convention, because the index i  appeared twice.)


The i is known as a dummy index since the result is not dependent on it; thus we could also write, for example:

v = vj ej.

An index that is not summed over is a free index and should be found in each term of the equation or formula.


In contexts where the index must appear once as a subscript and once as a superscript, the basis vectors ei retain subscripts but the coordinates become vi with superscripts. Then the basic rule is:

v = vi ei.

The value of the Einstein convention is that it applies to other vector spaces built from V  using the tensor product and duality. For example, Votimes V, the tensor product of V  with itself, has a basis consisting of tensors of the form mathbf{e}_{ij} = mathbf{e}_i otimes mathbf{e}_j. Any tensor T in Votimes V can be written as: In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

mathbf{T} = T^{ij}mathbf{e}_{ij}.

V*, the dual of V, has a basis e1, e2, ..., en which obeys the rule

mathbf{e}^i (mathbf{e}_j) = delta_{i}^j.

Here δ is the Kronecker delta, so delta_{i}^j is 1 if i =j  and 0 otherwise. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...


Examples

Einstein summation is clarified with the help of a few simple examples. Consider four-dimensional spacetime, where indices run from 0 to 3:

mathbf{} a^mu b_mu = a^0 b_0 + a^1 b_1 + a^2 b_2 + a^3 b_3
mathbf{} a^{munu} b_mu = a^{0nu} b_0 + a^{1nu} b_1 + a^{2nu} b_2 + a^{3nu} b_3.

The above example is one of contraction, a common tensor operation. The tensor mathbf{} a^{munu}b_{alpha} becomes a new tensor by summing over the first upper index and the lower index. Typically the resulting tensor is renamed with the contracted indices removed: In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ...

mathbf{} {s}^{nu} = a^{munu}b_{mu}.

For a familiar example, consider the dot product of two vectors a and b. The dot product is defined simply as summation over the indices of a and b:

mathbf{a}cdotmathbf{b} = a^{alpha}b_{alpha} = a^0 b_0 + a^1 b_1 + a^2 b_2 + a^3 b_3,

which is our familiar formula for the vector dot product. Remember it is sometimes necessary to change the components of a in order to lower its index; however, this is not necessary in Euclidean space, or any space with a metric equal to its inverse metric (e.g., flat spacetime). In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...


See also

Wikibooks
Wikibooks has a book on the topic of
General relativity:Einstein Summation Notation

Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... Penrose graphical notation is a visual representation of tensor equations in physics proposed by Roger Penrose. ... Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ...

References

  • Rawlings, Steve. "Lecture 10 - Einstein Summation Convention and Vector Identities", Oxford University, 2007-02-01.
  1. ^ Einstein, Albert (1916). "The Foundation of the General Theory of Relativity" (PDF). Annalen der Physik. Retrieved on 2006-09-03. 

  Results from FactBites:
 
Einstein notation - Wikipedia, the free encyclopedia (771 words)
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae.
Abstract index notation is an improvement of Einstein notation.
In Einstein notation, an index that is repeated twice in an equation implies a summation, and the summation symbol need not be included.
  More results at FactBites »

 
 

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