Editing Index notation (section)

==In mathematics==

{{Main|Ricci calculus|tensor}}

It is frequently helpful in mathematics to refer to the elements of an array using subscripts. The subscripts can be [[integer]]s or [[Variable (mathematics)|variables]]. The array takes the form of [[tensors]] in general, since these can be treated as multi-dimensional arrays. Special (and more familiar) cases are [[vector (geometry)|vectors]] (1d arrays) and [[matrix (mathematics)|matrices]] (2d arrays).

The following is only an introduction to the concept: index notation is used in more detail in mathematics (particularly in the representation and manipulation of [[tensor#Operations|tensor operations]]). See the main article for further details.

===One-dimensional arrays (vectors)===

{{main|Vector (mathematics and physics)}}

A vector treated as an array of numbers by writing as a [[row vector]] or [[column vector]] (whichever is used depends on convenience or context):

:<math>\mathbf{a} = \begin{pmatrix}
    a_1 \\
    a_2 \\
    \vdots \\
    a_n  
  \end{pmatrix}, \quad \mathbf{a} = \begin{pmatrix}
    a_1 & a_2 & \cdots & a_n
\end{pmatrix}</math>

Index notation allows indication of the elements of the array by simply writing ''a<sub>i</sub>'', where the index ''i'' is known to run from 1 to ''n'', because of n-dimensions.<ref>An introduction to Tensor Analysis: For Engineers and Applied Scientists, J.R. Tyldesley, Longman, 1975, {{ISBN|0-582-44355-5}}</ref>
For example, given the vector:

:<math>\mathbf{a} = \begin{pmatrix}
  10 & 8 & 9 & 6 & 3 & 5 \\ 
\end{pmatrix}</math>

then some entries are

:<math>a_1 = 10,\, a_2 = 8,\, \cdots,\, a_6 = 5 </math>.

The notation can be applied to [[vectors in mathematics and physics]]. The following [[vector equation]]

:<math>\mathbf{a} + \mathbf{b} = \mathbf{c}</math>

can also be written in terms of the elements of the vector (aka components), that is

:<math> a_i + b_i = c_i </math>

where the indices take a given range of values. This expression represents a set of equations, one for each index. If the vectors each have ''n'' elements, meaning ''i'' = 1,2,…''n'', then the equations are explicitly

:<math>\begin{align}
  a_1 + b_1 &= c_1 \\
  a_2 + b_2 &= c_2 \\
            &\ \ \vdots \\
  a_n + b_n &= c_n
\end{align}</math>

Hence, index notation serves as an efficient shorthand for
#representing the general structure to an equation,
#while applicable to individual components.

===Two-dimensional arrays===
[[File:Matrix.svg|thumb|247px|right|Elements of matrix '''A''' are described with two subscripts or indices.]]
{{main|matrix (mathematics)}}
{{see also|Dyadics}}

More than one index is used to describe arrays of numbers, in two or more dimensions, such as the elements of a matrix, (see also image to right);

:<math>\mathbf{A} = \begin{pmatrix}
  a_{11} & a_{12} & \cdots & a_{1n} \\
  a_{21} & a_{22} & \cdots & a_{2n} \\
  \vdots & \vdots & \ddots & \vdots \\
  a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{pmatrix}</math>

The entry of a matrix '''A''' is written using two indices, say ''i'' and ''j'', with or without commas to separate the indices: ''a<sub>ij</sub>'' or ''a<sub>i,j</sub>'', where the first subscript is the row number and the second is the column number. [[Multiplication|Juxtaposition]] is also used as notation for multiplication; this may be a source of confusion. For example, if

:<math>\mathbf{A} = \begin{pmatrix}
  9 & 8 & 6 \\
  1 & 2 & 7 \\
  4 & 9 & 2 \\
  6 & 0 & 5 
\end{pmatrix}</math>

then some entries are

:<math>a_{11} = 9,\, a_{12} = 8,\, a_{21} = 1,\, \cdots,\, a_{23} = 7,\, \cdots </math>.

For indices larger than 9, the comma-based notation may be preferable (e.g., ''a''<sub>3,12</sub> instead of ''a''<sub>312</sub>).

[[Matrix equation]]s are written similarly to vector equations, such as

:<math> \mathbf{A} + \mathbf{B} = \mathbf{C} </math>

in terms of the elements of the matrices (aka components)

:<math> A_{ij} + B_{ij} = C_{ij} </math>

for all values of ''i'' and ''j''. Again this expression represents a set of equations, one for each index. If the matrices each have ''m'' rows and ''n'' columns, meaning {{nowrap|''i'' {{=}} 1, 2, …, ''m''}} and {{nowrap|''j'' {{=}} 1, 2, …, ''n''}}, then there are ''mn'' equations.

===Multi-dimensional arrays===

{{main|Tensors}}
{{see also|Classical treatment of tensors}}

The notation allows a clear generalization to multi-dimensional arrays of elements: tensors. For example,

:<math> A_{i_1 i_2 \cdots } + B_{i_1 i_2 \cdots} = C_{i_1 i_2 \cdots} </math>

representing a set of many equations.

In tensor analysis, superscripts are used instead of subscripts to distinguish covariant from contravariant entities, see [[covariance and contravariance of vectors]] and [[raising and lowering indices]].