In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially the usage of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
in
Mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies
summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, mat ...
over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of
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. It is also the modern name for what used to be cal ...
; however, it is often used in physics applications that do not distinguish between
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
and
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. Typically, the cotangent space, T ...
s. It was introduced to physics by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
in 1916.
Introduction
Statement of convention
According to this convention, when an index variable appears twice in a single
term
Term may refer to:
* Terminology, or term, a noun or compound word used in a specific context, in particular:
**Technical term, part of the specialized vocabulary of a particular field, specifically:
***Scientific terminology, terms used by scient ...
and is not otherwise defined (see
Free and bound variables
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
), it implies summation of that term over all the values of the index. So where the indices can range over the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
,
:
is simplified by the convention to:
:
The upper indices are not
exponents
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
but are indices of coordinates,
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s or
basis vector
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s. That is, in this context should be understood as the second component of rather than the square of (this can occasionally lead to ambiguity). The upper index position in is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see ' below). Typically, would be equivalent to the traditional .
In
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 ...
, a common convention is that
* the
Greek alphabet
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as we ...
is used for space and time components, where indices take on values 0, 1, 2, or 3 (frequently used letters are ),
* the
Latin alphabet
The Latin alphabet or Roman alphabet is the collection of letters originally used by the ancient Romans to write the Latin language. Largely unaltered with the exception of extensions (such as diacritics), it used to write English and the o ...
is used for spatial components only, where indices take on values 1, 2, or 3 (frequently used letters are ),
In general, indices can range over any
indexing set, including an
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set th ...
. This should not be confused with a typographically similar convention used to distinguish between
tensor index notation
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. It is also the modern name for what used to be cal ...
and the closely related but distinct basis-independent
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. The indices are mere placeho ...
.
An index that is summed over is a ''summation index'', in this case "". It is also called a
dummy index since any symbol can replace "" without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term).
An index that is not summed over is a
''free index'' and should appear only once per term. If such an index does appear, it usually also appears in every other term in an equation. An example of a free index is the "" in the equation
, which is equivalent to the equation
.
Application
Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term.
When dealing with
covariant and contravariant vectors, where the position of an index also indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts; see ' below.
Vector representations
Superscripts and subscripts versus only subscripts
In terms of
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. In modern mathematical notation ...
,
* upper indices represent components of
contravariant vectors (
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
s),
* lower indices represent components of
covariant vectors (
covector
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
s).
They transform contravariantly or covariantly, respectively, with respect to
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. If two different bases are considere ...
.
In recognition of this fact, the following notation uses the same symbol both for a vector or covector and its ''components'', as in:
:
where is the vector and are its components (not the th covector ), is the covector and are its components. The basis vector elements
are each column vectors, and the covector basis elements
are each row covectors. (See also
#Abstract description;
duality
Duality may refer to:
Mathematics
* Duality (mathematics), a mathematical concept
** Dual (category theory), a formalization of mathematical duality
** Duality (optimization)
** Duality (order theory), a concept regarding binary relations
** Dual ...
, below and the
examples
Example may refer to:
* '' exempli gratia'' (e.g.), usually read out in English as "for example"
* .example, reserved as a domain name that may not be installed as a top-level domain of the Internet
** example.com, example.net, example.org, ex ...
)
In the presence of a
non-degenerate form (an
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. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
, for instance a
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
or
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
), one can
raise and lower indices.
A basis gives such a form (via the
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 th ...
), hence when working on with a
Euclidean metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore oc ...
and a fixed
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. For example, ...
, one has the option to work with only subscripts.
However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see
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. In modern mathematical notation ...
.
Mnemonics
In the above example, vectors are represented as
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
(column vectors), while covectors are represented as matrices (row covectors).
When using the column vector convention:
* "Upper indices go up to down; lower indices go left to right."
* "Covariant tensors are row vectors that have indices that are below (co-row-below)."
* Covectors are row vectors:
Hence the lower index indicates which ''column'' you are in.
* Contravariant vectors are column vectors:
Hence the upper index indicates which ''row'' you are in.
Abstract description
The virtue of Einstein notation is that it represents the
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
quantities with a simple notation.
In physics, a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
is invariant under transformations of basis. In particular, a
Lorentz scalar
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
is invariant under a
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. The respective inverse transformation i ...
. The individual terms in the sum are not. When the basis is changed, the ''components'' of a vector change by a
linear transformation
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 pre ...
described by a matrix. This led Einstein to propose the convention that repeated indices imply the summation is to be done.
As for covectors, they change by the
inverse matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.
The value of the Einstein convention is that it applies to other
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s built from using the
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 \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
and
duality
Duality may refer to:
Mathematics
* Duality (mathematics), a mathematical concept
** Dual (category theory), a formalization of mathematical duality
** Duality (optimization)
** Duality (order theory), a concept regarding binary relations
** Dual ...
. For example, , the tensor product of with itself, has a basis consisting of tensors of the form . Any tensor in can be written as:
:
, the dual of , has a basis , , …, which obeys the rule
:
where is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
. As
:
the row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.
Common operations in this notation
In Einstein notation, the usual element reference
for the
th row and
th column of matrix
becomes
. We can then write the following operations in Einstein notation as follows.
Inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
(hence also
vector dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
)
Using an
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. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis ...
, the inner product is the sum of corresponding components multiplied together:
:
This can also be calculated by multiplying the covector on the vector.
Vector 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 ...
Again using an orthogonal basis (in 3 dimensions) the cross product intrinsically involves summations over permutations of components:
:
where
:
is the
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 ...
, and is the generalized
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
. Based on this definition of , there is no difference between and but the position of indices.
Matrix-vector multiplication
The product of a matrix with a column vector is :
:
equivalent to
:
This is a special case of matrix multiplication.
Matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
The
matrix product
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
of two matrices and is:
:
equivalent to
:
Trace
Trace may refer to:
Arts and entertainment Music
* Trace (Son Volt album), ''Trace'' (Son Volt album), 1995
* Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* The Trace (album), ''The ...
For a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
, the trace is the sum of the diagonal elements, hence the sum over a common index .
Outer product
In linear algebra, the outer product of two 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. An ea ...
The outer product of the column vector by the row vector yields an matrix :
:
Since and represent two ''different'' indices, there is no summation and the indices are not eliminated by the multiplication.
Raising and lowering indices In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
Vectors, covectors and the metric
Math ...
Given a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
, one can raise an index or lower an index by contracting the tensor with the
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 ...
, . For example, take the tensor , one can raise an index:
:
Or one can lower an index:
:
See also
*
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
*
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. The indices are mere placeho ...
*
Bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathema ...
*
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. A diagram in the notation consists of several sha ...
*
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 ...
*
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. This involves functionals over the ''φs, func ...
Notes
#This applies only for numerical indices. The situation is the opposite for
abstract indices
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. The indices are mere placeho ...
. Then, vectors themselves carry upper abstract indices and covectors carry lower abstract indices, as per the example in the
introduction
Introduction, The Introduction, Intro, or The Intro may refer to:
General use
* Introduction (music), an opening section of a piece of music
* Introduction (writing), a beginning section to a book, article or essay which states its purpose and g ...
of this article. Elements of a basis of vectors may carry a lower ''numerical'' index and an upper ''abstract'' index.
References
Bibliography
* .
External links
*
*
{{tensors
Mathematical notation
Multilinear algebra
Tensors
Riemannian geometry
Mathematical physics
Albert Einstein