Index (mathematics)
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For other uses of "Index", see 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 transform.
- 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.
