Index (mathematics)

# Index (mathematics)

Discussion

Encyclopedia
The word index is used in variety of senses in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

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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. Example: In the list , the number is the index of the third element, . See also:
• Indexed family
Indexed family
In mathematics, an indexed family is a collection of values that are associated with indexes. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers....

- the list or array (e.g. )
• Index set
Index set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...

- The set of the indices themselves (e.g. )
• Index (information technology)
Index (information technology)
In computer science, an index can be:# an integer that identifies an array element# a data structure that enables sublinear-time lookup -Array element identifier:...

- a corresponding concept in information technology

## Algebra

• The word index sometimes denotes an exponent or the degree of an nth root
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

. An exponent index is generally written as a superscript, and an nth-root index as a small numeral within the radical symbol. Examples: In the expression , the index is , and the index is in the expression .

• The index of a subgroup
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...

is the number of its left coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...

s (which is equal to the number of its right cosets).

• The index of a linear transform, particularly a Fredholm operator
Fredholm operator
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....

, is the dimension of its kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...

minus the dimension of its cokernel
Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....

: see index of a linear transformation.

• The index of a real quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

Q is defined (but not always consistently) as pq where Q can be written as a difference of p squared linear terms and q squared linear terms.

## Analysis

• The index, or winding number
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...

, 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.

• The index of a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

v at an isolated zero is the degree
Degree of a continuous mapping
In topology, the degree is a numerical invariant that describes a continuous mapping between two compact oriented manifolds of the same dimension. Intuitively, the degree represents the number of times that the domain manifold wraps around the range manifold under the mapping...

of the map

taking points near the zero into the unit sphere
Unit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...

. This index is used in the statement of the Poincaré–Hopf theorem
Poincaré–Hopf theorem
In mathematics, the Poincaré–Hopf theorem is an important theorem that is still used today in differential topology...

which relates the sum of the indices of a vector field to the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

of the manifold. The hairy ball theorem
Hairy ball theorem
The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on an even-dimensional n-sphere. An ordinary sphere is a 2-sphere, so that this theorem will hold for an ordinary sphere...

is a special case. Confer fixed point index
Fixed point index
In mathematics, the fixed-point index is a concept in topological fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed points....

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