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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 Unit of angle 2 Degree of a polynomial 3 Degree of a field extension 4 Degree of a vertex in a graph 5 Degree of a continuous map 5.1 From a circle to itself 5.2 Between manifolds 5.3 Properties 6 Degree of freedom

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]

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.

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].

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.

Main article: degree (continuous map)

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

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

.

There is a projection

, ,

where [x] is the equivalence class of x modulo1 (i.e. if and only if x − y is an integer).

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

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.

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

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 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 .

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

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).

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