Index

The word index (plural: indices) 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.

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

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.