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写像度 브라우어르 차수 Degree of a continuous mapping Degré d'une application 映射度 Степень отображения Stopień Brouwera Brouwerscher Abbildungsgrad Grado topologico
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In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that 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. Le degré d'une application continue entre variétés de même dimension est une généralisation de la notion d'enroulement d'un cercle sur lui-même. C'est un invariant homologique à valeurs entières. Sa définition, d'abord réservée aux applications différentiables, s'étend aux applications continues par passage à la limite du fait de son invariance par homotopie. Mais la construction des groupes d'homologie permet aussi de proposer une définition directe pour les applications continues. 写像度（しゃぞうど、degree, mapping degree）とは、コンパクト、弧状連結、向き付けられた同次元の多様体間での連続写像を特徴付ける整数のこと。写像のホモトピー不変量のひとつである。 在拓扑学中，两个同维数流形之间的连续映射的度数（degree）非正式地说是一个点被盖住的次数。一个映射的度数可用同调群，或（对光滑映射）的原像定义。它是卷绕数的一个推广。例如，考虑复平面上映射 zn，视为 S2 到自身的映射，具有度数 n，它将球面绕自身缠了 n 圈。 在物理学中，连续映射的度数，比如从空间到有序参数集的一个映射，是的一个例子。 Степень отображения — гомотопический инвариант непрерывного отображения между компактными многообразиями равной размерности. В простейшем случае, для отображения из окружности в окружность степень отображения можно определить как число оборотов точки когда пробегает окружность. Stopień Brouwera lub inaczej stopień topologiczny – narzędzie pozwalające na określenie, czy dane równanie ma rozwiązanie. Jest jednym z niezmienników topologicznych i ma szerokie zastosowanie w nieliniowej analizie matematycznej. In matematica, e più precisamente in topologia, il grado topologico è una quantità introdotta da Luitzen Brouwer attorno al 1910 che misura il "numero di avvolgimento" di una funzione continua fra spazi topologici "della stessa dimensione". Questa quantità fornisce un'informazione sul comportamento qualitativo globale della funzione, ed è un invariante omotopico, cioè non cambia se la funzione viene deformata in modo continuo (una tale deformazione è chiamata omotopia). Il grado di una funzione viene solitamente indicato con deg . 대수적 위상수학에서, 두 다양체 사이의 연속 함수의 브라우어르 차수(Brouwer次數, Brouwer degree)는 함수의 정의역이 함수의 치역을 몇 번 감싸는지를 나타내는 정수이다. 기호는 .
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대수적 위상수학에서, 두 다양체 사이의 연속 함수의 브라우어르 차수(Brouwer次數, Brouwer degree)는 함수의 정의역이 함수의 치역을 몇 번 감싸는지를 나타내는 정수이다. 기호는 . Stopień Brouwera lub inaczej stopień topologiczny – narzędzie pozwalające na określenie, czy dane równanie ma rozwiązanie. Jest jednym z niezmienników topologicznych i ma szerokie zastosowanie w nieliniowej analizie matematycznej. 在拓扑学中，两个同维数流形之间的连续映射的度数（degree）非正式地说是一个点被盖住的次数。一个映射的度数可用同调群，或（对光滑映射）的原像定义。它是卷绕数的一个推广。例如，考虑复平面上映射 zn，视为 S2 到自身的映射，具有度数 n，它将球面绕自身缠了 n 圈。 在物理学中，连续映射的度数，比如从空间到有序参数集的一个映射，是的一个例子。 In matematica, e più precisamente in topologia, il grado topologico è una quantità introdotta da Luitzen Brouwer attorno al 1910 che misura il "numero di avvolgimento" di una funzione continua fra spazi topologici "della stessa dimensione". Questa quantità fornisce un'informazione sul comportamento qualitativo globale della funzione, ed è un invariante omotopico, cioè non cambia se la funzione viene deformata in modo continuo (una tale deformazione è chiamata omotopia). L'esempio fondamentale è quello di una funzione continua tra due circonferenze: il grado topologico è il "numero di avvolgimenti" che la funzione fa compiere alla circonferenza. Il grado di una funzione viene solitamente indicato con deg . Le degré d'une application continue entre variétés de même dimension est une généralisation de la notion d'enroulement d'un cercle sur lui-même. C'est un invariant homologique à valeurs entières. Sa définition, d'abord réservée aux applications différentiables, s'étend aux applications continues par passage à la limite du fait de son invariance par homotopie. Mais la construction des groupes d'homologie permet aussi de proposer une définition directe pour les applications continues. Степень отображения — гомотопический инвариант непрерывного отображения между компактными многообразиями равной размерности. В простейшем случае, для отображения из окружности в окружность степень отображения можно определить как число оборотов точки когда пробегает окружность. In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that 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 homotopy invariant (invariant among homotopies), 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. 写像度（しゃぞうど、degree, mapping degree）とは、コンパクト、弧状連結、向き付けられた同次元の多様体間での連続写像を特徴付ける整数のこと。写像のホモトピー不変量のひとつである。
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