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Degree (mathematics)

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

 Contents

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 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

From a circle to itself

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

$f:S^1\to S^1 \,$.

There is a projection

$R \to S^1=R/Z \,$, $x\mapsto [x]$,

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

If $f : S^1 \to S^1$ is continuous then there exists a continuous $F : R \to R$, called a lift of f to 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

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

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

fm([X]) = deg(f)[Y].

Here fm 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) = {x1,x2,..,xn} 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) = {x1,x2,..,xn} 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 the article "Degrees of freedom" for the use of this concept.

03-10-2013 05:06:04