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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

1 Unit of angle
2 Degree of a polynomial
3 Degree of an algebraic number
4 Degree of a field extension
5 Degree of a vertex in a graph
6 Degree of a continuous map
7 Degree of freedom
8 References

Unit of angle

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. This article describes the unit of angle For other meanings see Degree. The degree symbol (° Unicode: U+00B0 HTML: &deg is a typographical symbol or Glyph, that is used to represent degrees of arc (see In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called 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. This article is about the geographical concept For other uses of the word see Meridian. A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe A geographic coordinate system enables every location on the Earth to be specified in three coordinates using mainly a spherical coordinate system. In Astronomy and Navigation, the celestial sphere is an imaginary rotating Sphere of "gigantic Radius " ^[1]

Degree of a polynomial

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. When a Polynomial is expressed as a sum or difference of terms (e In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations 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 an algebraic number

The degree of an algebraic number is the smallest degree of a non-trivial polynomial in one variable with rational coefficients having said algebraic number as a root. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or For instance, any rational number $q$ is degree 1 since it is the root of the polynomial $x\mapsto x-q$ .

Additionally, the square root of any non-square positive integer, say $\sqrt n$ , is degree 2, as it is the root of $x\mapsto x^2-n$ .

Degree of a field extension

Main article: field extension

Given a field extension K/F, the field K can be considered as a vector space over the field F. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The dimension of this vector space is the degree of the extension and is denoted by [K : F]. In Mathematics, the dimension of a Vector space V is the cardinality (i

Degree of a vertex in a graph

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 Graph theory, the degree (or valency) of a vertex of a graph is the number of edges incident to the vertex In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In a directed graph, the indegree and outdegree count the number of directed edges coming into and out of a vertex respectively. In Mathematics and Computer science, a graph is the basic object of study in Graph theory.

Degree of a continuous map

Main article: degree (continuous map)

In topology, the term degree is applied to continuous maps between manifolds of the same dimension. This article is about the term "degree" as used in algebraic topology Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

From a circle to itself

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

$f\colon S^1\to S^1 \,$ . In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

There is a projection

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

where $[x]$ is the equivalence class of $x$ modulo1 (i. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure 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 $f([z]) = [F(z)] \,$ . 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

Let $f:X\to Y \,$ be a continuous map, $X$ and $Y$ closed oriented $m$ -dimensional manifolds. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Then the degree of $f$ is an integer such that

$f_m([X])=\deg(f)[Y]. \,$

Here $f m$ is the map induced on the $m$ dimensional homology group, $[X]$ and $[Y]$ denote the fundamental classes of $X$ and $Y$ . In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, the fundamental class is a homology class associated to an Oriented Manifold M, which corresponds to "the whole

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. Topological equivalence redirects here see also Topological equivalence (dynamical systems. 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 $Y$ . For a different notion of boundary related to Manifolds see that article

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. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and See degrees of freedom.

References

^ Beckmann P. (1976) A History of Pi, St. Martin's Griffin. ISBN 0-312-38185-9

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