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indices

In perhaps the most frequent sense, an index is a superscriptor subscriptto a symbol. Superscript indices are often, but not always, used to indicate powers. Subscript indices are usually used to identify an element of a setor arrayor 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 subgroupis 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 operatoris the dimension of its kernelminus the dimension of its cokernel: see index of a linear transformation. The index of a real quadratic formQ 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 fieldv at an isolated zero is the degreeof the map

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