Index (Mathematics) Information
The word index (plural: indices) is used in variety of senses in mathematics.
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General
- In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a subscript to the variable. [1] Example: In the list {x1,x2,x3,x4}, the number 3 is the index of the third element, x3. See also:
- Indexed family - the list or array (e.g. {x1,x2,x3,x4})
- Index set - The set of the indices themselves (e.g. {1,2,3,4})
- Index (information technology) - a corresponding concept in information technology
Algebra
- The word index sometimes denotes an exponent or the degree of an nth root. An exponent index is generally written as a superscript, and an nth-root index as a small numeral within the radical symbol.[1] Examples: In the expression 25, the index is 5, and the index is 1 / 3 in the expression .
- The index of a subgroup is the number of its left cosets (which is equal to the number of its right cosets). [1]
- 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.
Analysis
- The index, or winding number, of an oriented closed curve on a surface relative to a point on that surface is (loosely speaking) the number of times the curve goes around the point counter-clockwise. [1]
- 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.
References
- ^ a b c d Borowski and Borwein, Collins Dictionary of Mathematics, second edition, HarperCollins Publishers, London, 1989, 2002. p 277.
Categories: Mathematical terminology
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