Degree (mathematics)

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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 1 Degree of a polynomial 2 Degree of a field extension 3 Degree of a vertex in a graph 4 Degree of a continuous map 4.1 From a circle to itself 4.2 Between manifolds 4.3 Properties 5 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 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 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.

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

<math>f\colon S^1\to S^1 \,</math>.

There is a projection

<math>\mathbb R \to S^1= \mathbb R/ \mathbb Z \,</math>, <math>x\mapsto [x]</math>,

where <math>[x]</math> is the equivalence class of <math>x</math> modulo 1 (i.e. <math>x\sim y</math> iff <math>x-y</math> is an integer).

If <math>f : S^1 \to S^1 \,</math> is continuous then there exists a continuous <math>F : \mathbb R \to \mathbb R</math>, called a lift of <math>f</math> to <math>\mathbb R</math>, such that <math>f([z]) = [F(z)] \,</math>. Such a lift is unique up to an additive integer constant and <math>deg(f)= F(x + 1)-F(x) \,</math>.

Note that <math>F(x + 1)-F(x)</math> is an integer and it is also continuous with respect to <math>x</math>; therefore the definition does not depend on choice of <math>x</math>.

Between manifolds

Let <math>f:X\to Y \,</math> be a continuous map, <math>X</math> and <math>Y</math> closed oriented <math>m</math>-dimensional manifolds. Then the degree of <math>f</math> is an integer such that

<math>f_m([X])=\deg(f)[Y]. \,</math>

Here <math>f_m</math> is the map induced on the <math>m</math> dimensional homology group, <math>[X]</math> and <math>[Y]</math> denote the fundamental classes of <math>X</math> and <math>Y</math>.

Here is the easiest way to calculate the degree: If <math>f</math> is smooth and <math>p</math> is a regular value of <math>f</math> then <math>f^{-1}(p)=\{x_1,x_2,..,x_n\} \,</math> is a finite number of points. In a neighborhood of each the map <math>f</math> is a homeomorphism to its image, so it might be orientation preserving or orientation reversing. If <math>m</math> is the number of orientation preserving and <math>k</math> is the number of orientation reversing locations, then <math>deg(f)=m-k \,</math>.

The same definition works for compact manifolds with boundary but then <math>f</math> should send the boundary of <math>X</math> to the boundary of <math>Y</math>.

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 <math>f^{-1}(p)=\{x_1,x_2,..,x_n\} \,</math> 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 <math>f,g:S^n\to S^n \,</math> 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.eo:Vikipedio:Projekto matematiko/Grado (matematiko)