About: Symbol of a differential operator

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In mathematics, the symbol of a linear differential operator is a polynomial representing a differential operator, which is obtained, roughly speaking, by replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this connection it leads to the notion of a pseudo-differential operator. The highest-order terms of the symbol, known as the principal symbol, almost completely controls the qualitative behavior of solutions of a partial differential equation. Linear elliptic partial differential equations can be characterized as those whose principal symbol is nowhere zero. In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the character

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rdfs:label	Symbole d'un opérateur différentiel (fr) 微分作用素の表象 (ja) 주표상 (ko) Symbol of a differential operator (en)
rdfs:comment	数学の分野における微分作用素の表象（びぶんさようそのひょうしょう、英: symbol of a differential operator）とは、大雑把に言うと、各偏微分を新たな変数に置き換えることによって、微分作用素を多項式へと関連付けるものである。フーリエ解析の分野において幅広く用いられている。特に、擬微分作用素の概念は、この表象の関連付けにより導かれるものである。表象の内、最高次のものは主表象 (principal symbol) と呼ばれ、偏微分方程式の解の定性的な挙動をほぼ完全に決定付けるものである。線型の楕円型偏微分方程式は、主表象が至る所零とならないようなものとして特徴付けられる。双曲型偏微分方程式と放物型偏微分方程式の研究においては、主表象の零点は偏微分方程式の特性超曲面と対応する。したがって、表象はそれらの方程式の解に関する重要な概念であり、それらの解の特異性を調べる上で用いられる主要な道具の内の一つである。 (ja) En mathématiques, le symbole d'un opérateur différentiel est le polynôme obtenu à partir d'un opérateur différentiel linéaire en remplaçant, grosso modo, chaque dérivée partielle par une indéterminée. Le symbole d'un opérateur différentiel a d'importantes applications en analyse de Fourier puisqu'il représente l'effet de l'opérateur sur le spectre d'une fonction. Pris en sens inverse, ce lien conduit à une notion plus générale d'opérateur pseudo-différentiel. (fr) In mathematics, the symbol of a linear differential operator is a polynomial representing a differential operator, which is obtained, roughly speaking, by replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this connection it leads to the notion of a pseudo-differential operator. The highest-order terms of the symbol, known as the principal symbol, almost completely controls the qualitative behavior of solutions of a partial differential equation. Linear elliptic partial differential equations can be characterized as those whose principal symbol is nowhere zero. In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the character (en)
dcterms:subject	Vector bundles Differential operators
Wikipage page ID	10639143 (xsd:integer)
Wikipage revision ID	1087294886 (xsd:integer)
Link from a Wikipage to another Wikipage	Pseudo-differential operator Multiplier (Fourier analysis) Hyperbolic partial differential equation Mathematics Elliptic partial differential equation Closed manifold Kernel (algebra) Parabolic partial differential equation Partial derivative Vector bundles Local coordinates Euclidean space Fourier analysis Fourier transform Partial differential equation Differential operator Atiyah–Singer index theorem Differential operators Cotangent bundle Tensor Tensor product Symmetric power Homogeneous polynomial Symmetric tensor Bundle map Polynomial Fredholm operator Schwartz function Method of characteristics Vector bundle Fourier multiplier Elliptic differential operator Compact manifold Multi-index
sameAs	Symbol of a differential operator Symbol of a differential operator Symbol of a differential operator Symbol of a differential operator Symbol of a differential operator Symbol of a differential operator Symbol of a differential operator
dbp:wikiPageUsesTemplate	dbt:Citation dbt:No_footnotes dbt:Reflist dbt:Short_description
has abstract	In mathematics, the symbol of a linear differential operator is a polynomial representing a differential operator, which is obtained, roughly speaking, by replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this connection it leads to the notion of a pseudo-differential operator. The highest-order terms of the symbol, known as the principal symbol, almost completely controls the qualitative behavior of solutions of a partial differential equation. Linear elliptic partial differential equations can be characterized as those whose principal symbol is nowhere zero. In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the characteristics of the partial differential equation. Consequently, the symbol is often fundamental for the solution of such equations, and is one of the main computational devices used to study their singularities. (en) En mathématiques, le symbole d'un opérateur différentiel est le polynôme obtenu à partir d'un opérateur différentiel linéaire en remplaçant, grosso modo, chaque dérivée partielle par une indéterminée. Le symbole d'un opérateur différentiel a d'importantes applications en analyse de Fourier puisqu'il représente l'effet de l'opérateur sur le spectre d'une fonction. Pris en sens inverse, ce lien conduit à une notion plus générale d'opérateur pseudo-différentiel. Le terme de plus haut degré du symbole, appelé symbole principal, contrôle presque complètement le comportement qualitatif des solutions de l'équation aux dérivées partielles linéaire correspondante à l'opérateur différentiel. Les équations aux dérivées partielles elliptiques peuvent être caractérisées comme celles dont le symbole principal correspondant ne s'annule jamais. Dans l'étude des équation aux dérivées partielles hyperboliques, les zéros du symbole principal correspondent aux caractéristiques de l'équation. La notion d'opérateur différentiel s'étend au cadre des variétés, mais les coefficients sont modifiés lors des changements de cartes. On arrive cependant à définir le symbole principal sous forme d'un tenseur symétrique, ce qui permet de retrouver les concepts d'opérateur pseudo-différentiels, d'opérateur elliptique... (fr) 数学の分野における微分作用素の表象（びぶんさようそのひょうしょう、英: symbol of a differential operator）とは、大雑把に言うと、各偏微分を新たな変数に置き換えることによって、微分作用素を多項式へと関連付けるものである。フーリエ解析の分野において幅広く用いられている。特に、擬微分作用素の概念は、この表象の関連付けにより導かれるものである。表象の内、最高次のものは主表象 (principal symbol) と呼ばれ、偏微分方程式の解の定性的な挙動をほぼ完全に決定付けるものである。線型の楕円型偏微分方程式は、主表象が至る所零とならないようなものとして特徴付けられる。双曲型偏微分方程式と放物型偏微分方程式の研究においては、主表象の零点は偏微分方程式の特性超曲面と対応する。したがって、表象はそれらの方程式の解に関する重要な概念であり、それらの解の特異性を調べる上で用いられる主要な道具の内の一つである。 (ja)
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page length (characters) of wiki page	5699 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Symbol_of_a_differential_operator
is Link from a Wikipage to another Wikipage of	D-module Chern–Gauss–Bonnet theorem Elliptic complex Elliptic operator General Leibniz rule Oscillatory integral operator GJMS operator Fourier integral operator Atiyah–Singer index theorem Paneitz operator Holmgren's uniqueness theorem Homogeneous polynomial Method of characteristics Noncommutative algebraic geometry Principal symbol
is Wikipage redirect of	Principal symbol
is foaf:primaryTopic of	wikipedia-en:Symbol_of_a_differential_operator

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