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.

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 : 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

f\colon S^1\to S^1 \,

There is a projection

\mathbb R \to S^1= \mathbb R/ \mathbb Z \,

x\mapsto

^*,

where ^* is the equivalence class of $x$ modulo1 (i.e. $x\sim y$ if and only if $x-y$ is an integer).

If $f : S^1 \to S^1 \,$ is continuous then there exists a continuous $F : \mathbb R \to \mathbb R$ , called a lift of $f$ to $\mathbb R$ , such that $f(= [F(z) \,$ . 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 $f:X\to Y \,$ be a continuous map, $X$ and $Y$ closed oriented $m$ -dimensional manifolds. Then the degree of $f$ is an integer such that

f_m(

^*)=\deg(f)^*. \,

Here $f_m$ is the map induced on the $m$ dimensional homology group, $and [Y$ 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 Z₂ homology. In this case deg₂(f) is element of Z₂, 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 $f,g:S^n\to S^n \,$ 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.

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