Index (mathematics)

The word "index" is used in variety of senses in mathematics.

* In perhaps the most frequent sense, an index is a superscript or subscript to a symbol. Superscript indices are often, but not always, used to indicate powers. Subscript indices are usually used to identify an element of a set or array or sequence of variables. See also "index set", "indexed family", and Index (information technology).

* Index meaning exponent, the power to which some base is raised. See also 'Indices Laws'/'Laws of Indices'/'Laws of Exponents'.

* The index of a subgroup is the number of its left cosets (which is equal to the number of its right cosets).

* The index of a Fredholm operator is the dimension of its kernel minus the dimension of its cokernel.

* 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 of a vector field "v" at an isolated zero is the degree of the map

::$x^a\; mapsto\; frac\{v^a(x)\}\{sqrt\{sum\_b(v^b(x))^2$

: 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.

* "An index relates the value of a variable (or group of variables) to a base level, which is often the value on a particular date. The base level is set so that the index produces numbers that are easy to understand and compare. Indices are used to report on a wide variety of variables, including prices and wages, ultraviolet levels in sunlight, and even the readability of textbooks." from Mathematics of Data Management published by McGraw-Hill Ryerson