# Degree (mathematics)

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

## Contents

## 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 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

- .

There is a projection

- , ,

where is the equivalence class of modulo 1 (i.e. iff is an integer).

If is continuous then there exists a continuous , called a *lift* of to , such that . Such a lift is unique up to an additive integer constant and .

Note that is an integer and it is also continuous with respect to ; therefore the definition does not depend on choice of .

### Between manifolds

Let be a continuous map, and closed oriented -dimensional manifolds.
Then the **degree** of is an integer such that

Here is the map induced on the dimensional homology group, and denote the fundamental classes of and .

Here is the easiest way to calculate the degree: If is smooth and is a regular value of then is a finite number of points. In a neighborhood of each the map is a homeomorphism to its image, so it might be orientation preserving or orientation reversing. If is the number of orientation preserving and is the number of orientation reversing locations, then .

The same definition works for compact manifolds with boundary but then should send the boundary of to the boundary of .

One can also define **degree modulo 2** (deg_{2}(*f*)) the same way as before but taking the *fundamental class* in **Z**_{2} homology. In this case deg_{2}(*f*) is element of **Z**_{2}, the manifolds need not be orientable and if as before then deg_{2}(*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 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.