**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:

“It is true that genius takes its rise out of the mountains of rectitude; that all beauty and power which men covet are somehow born out of that Alpine district; that any extraordinary *degree of* beauty in man or woman involves a moral charm.”

—Ralph Waldo Emerson (1803–1882)

“Essential truth, the truth of the intellectualists, the truth with no one thinking it, is like the coat that fits tho no one has ever tried it on, like the music that no ear has listened to. It is less real, not more real, than the verified article; and to attribute a superior *degree* of glory to it seems little more than a piece of perverse abstraction-worship.”

—William James (1842–1910)

“The gap between ideals and actualities, between dreams and achievements, the gap that can spur strong men to increased exertions, but can break the spirit of others—this gap is the most conspicuous, *continuous* land mark in American history. It is conspicuous and *continuous* not because Americans achieve little, but because they dream grandly. The gap is a standing reproach to Americans; but it marks them off as a special and singularly admirable community among the world’s peoples.”

—George F. Will (b. 1941)