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Index (mathematics)

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In mathematics, 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 (mathematics)|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 (algebra) minus the dimension of its kernel (algebra) 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 a continuous mapping|degree of the map :

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