Degree (mathematics)

This article is about the term "degree" as used in mathematics. For alternate meanings, see degree.

In mathematics, there are several meanings of degree depending on the subject.

Unit of angle

Main article: Degree (angle)

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of plane angle, representing ¹⁄₃₆₀ of a full rotation. When that angle is with respect to a reference meridian, it indicates a location along a great circle of a sphere, such as Earth (see Geographic coordinate system), Mars, or the celestial sphere.^[1]

Degree of a polynomial

Main article: Degree of a polynomial

The degree of a term of a polynomial in one variable is the exponent on the variable in that term; the degree of a polynomial is the highest such degree. For example, in 2x³ + 4x² + x + 7, the term of highest degree is 2x³; this term, and therefore the entire polynomial, are said to have degree 3.

For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree of the polynomial is again the highest such degree. For example, the polynomial x²y² + 3x³ + 4y has degree 4, the same degree as the term x²y².

Degree of an algebraic number

The degree of an algebraic number is the smallest degree of a non-trivial polynomial in one variable with rational coefficients having said algebraic number as a root. For instance, any rational number Failed to parse (Missing texvc executable; please see math/README to configure.): q

is degree 1 since it is the root of the polynomial Failed to parse (Missing texvc executable; please see math/README to configure.): x\mapsto x-q

.

Additionally, the square root of any non-square positive integer, say Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt n , is degree 2, as it is the root of Failed to parse (Missing texvc executable; please see math/README to configure.): x\mapsto x^2-n .

Degree of a field extension

Main article: field extension

Given a field extension K/F, the field K can be considered as a vector space over the field F. The dimension of this vector space is the degree of the extension and is denoted by [K : F].

Degree of a vertex in a graph

Main article: degree (graph theory)

In graph theory, the degree of a vertex in a graph is the number of edges incident to that vertex — in other words, the number of lines coming out of the point. In a directed graph, the indegree and outdegree count the number of directed edges coming into and out of a vertex respectively.

Degree of a continuous map

Main article: degree (continuous map)

In topology, the term degree is applied to continuous maps between manifolds of the same dimension.

From a circle to itself

The simplest and most important case is the degree of a continuous map

Failed to parse (Missing texvc executable; please see math/README to configure.): f\colon S^1\to S^1 \,

.

There is a projection

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb R \to S^1= \mathbb R/ \mathbb Z \,

, Failed to parse (Missing texvc executable; please see math/README to configure.): x\mapsto [x] ,

where Failed to parse (Missing texvc executable; please see math/README to configure.): [x]

is the equivalence class of Failed to parse (Missing texvc executable; please see math/README to configure.): x
modulo1 (i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): x\sim y
if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): x-y
is an integer). 

If Failed to parse (Missing texvc executable; please see math/README to configure.): f : S^1 \to S^1 \,

is continuous then there exists a continuous Failed to parse (Missing texvc executable; please see math/README to configure.): F : \mathbb R \to \mathbb R

, called a lift of Failed to parse (Missing texvc executable; please see math/README to configure.): f

to Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb R

, such that Failed to parse (Missing texvc executable; please see math/README to configure.): f([z]) = [F(z)] \, . Such a lift is unique up to an additive integer constant and Failed to parse (Missing texvc executable; please see math/README to configure.): deg(f)= F(x + 1)-F(x) \, .

Note that Failed to parse (Missing texvc executable; please see math/README to configure.): F(x + 1)-F(x)

is an integer and it is also continuous with respect to Failed to parse (Missing texvc executable; please see math/README to configure.): x

therefore the definition does not depend on choice of Failed to parse (Missing texvc executable; please see math/README to configure.): x

.

Between manifolds

Let Failed to parse (Missing texvc executable; please see math/README to configure.): f:X\to Y \,

be a continuous map, Failed to parse (Missing texvc executable; please see math/README to configure.): X
and Failed to parse (Missing texvc executable; please see math/README to configure.): Y
closed oriented Failed to parse (Missing texvc executable; please see math/README to configure.): m

-dimensional manifolds. Then the degree of Failed to parse (Missing texvc executable; please see math/README to configure.): f

is an integer such that 

Failed to parse (Missing texvc executable; please see math/README to configure.): f_m([X])=\deg(f)[Y]. \,

Here Failed to parse (Missing texvc executable; please see math/README to configure.): f_m

is the map induced on the Failed to parse (Missing texvc executable; please see math/README to configure.): m
dimensional homology group, Failed to parse (Missing texvc executable; please see math/README to configure.): [X]
and Failed to parse (Missing texvc executable; please see math/README to configure.): [Y]
denote the fundamental classes of Failed to parse (Missing texvc executable; please see math/README to configure.): X
and Failed to parse (Missing texvc executable; please see math/README to configure.): Y

.

Here is the easiest way to calculate the degree: If Failed to parse (Missing texvc executable; please see math/README to configure.): f

is smooth and Failed to parse (Missing texvc executable; please see math/README to configure.): p
is a regular value of Failed to parse (Missing texvc executable; please see math/README to configure.): f
then Failed to parse (Missing texvc executable; please see math/README to configure.): f^{-1}(p)=\{x_1,x_2,..,x_n\} \,
is a finite number of points. In a neighborhood of each the map Failed to parse (Missing texvc executable; please see math/README to configure.): f
is a homeomorphism to its image, so it might be orientation preserving or orientation reversing. If Failed to parse (Missing texvc executable; please see math/README to configure.): m
is the number of orientation preserving and Failed to parse (Missing texvc executable; please see math/README to configure.): k
is the number of orientation reversing locations, then Failed to parse (Missing texvc executable; please see math/README to configure.): deg(f)=m-k \,

.

The same definition works for compact manifolds with boundary but then Failed to parse (Missing texvc executable; please see math/README to configure.): f

should send the boundary of Failed to parse (Missing texvc executable; please see math/README to configure.): X
to the boundary of Failed to parse (Missing texvc executable; please see math/README to configure.): Y

.

One can also define degree modulo 2 (deg₂(f)) the same way as before but taking the fundamental class in Z₂ homology. In this case deg₂(f) is element of Z₂, the manifolds need not be orientable and if Failed to parse (Missing texvc executable; please see math/README to configure.): f^{-1}(p)=\{x_1,x_2,..,x_n\} \,

as before then deg2(f) is n modulo 2.

Properties

The degree of map is a homotopy invariant; moreover for continuous maps from the sphere to itself it is a complete homotopy invariant, i.e. two maps Failed to parse (Missing texvc executable; please see math/README to configure.): f,g:S^n\to S^n \,

are homotopic if and only if deg(f) = deg(g).

Degree of freedom

A degree of freedom is a concept in mathematics, statistics, physics and engineering. See degrees of freedom.

References

1. ^ Beckmann P. (1976) A History of Pi, St. Martin's Griffin. ISBN 0-312-38185-9

es:Grado (matemática) fr:Degré (mathématiques) qu:K'atma