Degree of A Continuous Mapping

Degree Of A Continuous Mapping

In topology, the degree is a numerical invariant that describes a continuous mapping between two compact oriented manifolds of the same dimension. Intuitively, the degree represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations.

The degree of a map was first defined by Brouwer, who showed that the degree is a homotopy invariant, and used it to prove the Brouwer fixed point theorem. In modern mathematics, the degree of a map plays an important role in topology and geometry. In physics, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a topological quantum number.

Read more about Degree Of A Continuous Mapping:  Properties

Other articles related to "degree of a continuous mapping, degree of a, continuous, degree":

Degree Of A Continuous Mapping - Properties
... The degree of a map is a homotopy invariant moreover for continuous maps from the sphere to itself it is a complete homotopy invariant, i.e ... In other words, degree is an isomorphism ...

Famous quotes containing the words degree of, degree and/or continuous:

    Nothing in life possesses value except the degree of power—assuming that life itself is the will to power.
    Friedrich Nietzsche (1844–1900)

    In all pointed sentences, some degree of accuracy must be sacrificed to conciseness.
    Samuel Johnson (1709–1784)

    If an irreducible distinction between theatre and cinema does exist, it may be this: Theatre is confined to a logical or continuous use of space. Cinema ... has access to an alogical or discontinuous use of space.
    Susan Sontag (b. 1933)