Index (mathematics)

The word index (plural: indices) is used in variety of senses in mathematics.

General

In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a subscript to the variable. ^[1] Example: In the list , the number is the index of the third element, . See also:
- Indexed family - the list or array (e.g. $\lbrace x_1, x_2, x_3, x_4\rbrace$ )
- Index set - The set of the indices themselves (e.g. $\lbrace 1, 2, 3, 4\rbrace$ )
- Index (information technology) - a corresponding concept in information technology

The word index sometimes denotes an exponent or the degree of an nth root. An exponent index is generally written as a superscript, and an nth-root index as a small numeral within the radical symbol.^[1] Examples: In the expression $2^5$ , the index is $5$ , and the index is $1/3$ in the expression $\sqrt[3]{8}$ .

The index of a subgroup is the number of its left cosets (which is equal to the number of its right cosets). ^[1]

The index of a linear transform, particularly a Fredholm operator, is the dimension of its kernel minus the dimension of its cokernel: see index of a linear transformation.

The index of a real quadratic form Q is defined (but not always consistently) as p − q where Q can be written as a difference of p squared linear terms and q squared linear terms.

The index, or winding number, of an oriented closed curve on a surface relative to a point on that surface is (loosely speaking) the number of times the curve goes around the point counter-clockwise. ^[1]

$x^a \mapsto \frac{v^a(x)}{\sqrt{\sum_b(v^b(x))^2}}$

taking points near the zero into the unit sphere. This index is used in the statement of the Poincaré–Hopf theorem which relates the sum of the indices of a vector field to the Euler characteristic of the manifold. The hairy ball theorem is a special case. Confer fixed point index.

^ ^a ^b ^c ^d Borowski and Borwein, Collins Dictionary of Mathematics, second edition, HarperCollins Publishers, London, 1989, 2002. p 277.