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.

Contents

Degree of a polynomial

Degree of a field extension

Degree of a vertex in a graph

Degree of a continuous map

From a circle to itself

Between manifolds

Properties

Degree of freedom

Degree of a polynomial

:''See 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 2''x''³ + 4''x''² + ''x'' + 7, the term of highest degree is 2''x''³; 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 ''x''²''y''² + 3''x''³ + 4''y'' has degree 4, the same degree as the term ''x''²''y''².

Degree of a field extension

:''See 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

:''See 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

:''See 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
:$fcolon\; S^1\; o\; S^1\; ,$.
There is a projection
:$mathbb\; R\; o\; S^1=\; mathbb\; R/\; mathbb\; Z\; ,$, $xmapsto\; [x]$,
where $[x]$ is the equivalence class of $x$ modulo1 (i.e. $xsim\; y$ if and only if $x-y$ is an integer).
If $f\; :\; S^1\; o\; S^1\; ,$ is continuous then there exists a continuous $F\; :\; mathbb\; R\; o\; mathbb\; R$, called a ''lift'' of $f$ to $mathbb\; R$, such that $f([z])\; =\; [F(z)]\; ,$. Such a lift is unique up to an additive integer constant and $deg(f)=\; F(x\; +\; 1)-F(x)\; ,$.
Note that $F(x\; +\; 1)-F(x)$ is an integer and it is also continuous with respect to $x$; therefore the definition does not depend on choice of $x$.

Between manifolds

Let $f:X\; o\; Y\; ,$ be a continuous map, $X$ and $Y$ closed oriented $m$-dimensional manifolds.
Then the 'degree' of $f$ is an integer such that
:$f\_m([X])=deg(f)[Y].\; ,$
Here $f\_m$ is the map induced on the $m$ dimensional homology group, $[X]$ and $[Y]$ denote the fundamental classes of $X$ and $Y$.
Here is the easiest way to calculate the degree: If $f$ is smooth and $p$ is a regular value of $f$ then $f^\{-1\}(p)=\{x\_1,x\_2,..,x\_n\}\; ,$ is a finite number of points. In a neighborhood of each the map $f$ is a homeomorphism to its image, so it might be orientation preserving or orientation reversing. If $m$ is the number of orientation preserving and $k$ is the number of orientation reversing locations, then $deg(f)=m-k\; ,$.
The same definition works for compact manifolds with boundary but then $f$ should send the boundary of $X$ to the boundary of $Y$.
One can also define 'degree modulo 2' (deg₂(''f'')) the same way as before but taking the ''fundamental class'' in 'Z'₂ homology. In this case deg₂(''f'') is element of 'Z'₂, the manifolds need not be orientable and if $f^\{-1\}(p)=\{x\_1,x\_2,..,x\_n\}\; ,$ 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 $f,g:S^n\; o\; S^n\; ,$ 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.