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 label a set or sequence of variables. See also index set and indexed family.

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.

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

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