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).

★ 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 rac{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

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