Queries, comments, suggestions? Please contact us.



Area Mathematics - General concepts and linear algebra / Vectors and tensors

IEV ref102-03-41

en
dyadic product
tensor product, <of two vectors>
for two vectors U and V in an n-dimensional Euclidean space, tensor of the second order defined by the bilinear form f(X,Y)=(UX)(VY) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC iwaiaabYcacaaMe8UaaCywaiaacMcacqGH9aqpcaGGOaGaaCyvaiab gwSixlaahIfacaGGPaGaaiikaiaahAfacqGHflY1caWHzbGaaiykaa aa@4ADC@ , where X and Y are any vectors in the same space

Note 1 to entry: The bilinear form can be represented by f( X,Y )=( i U i X i )( j V j Y j )= ij U i V j X i Y j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaamOzaOWaae WaaeaacaWHybGaaiilaiaahMfaaiaawIcacaGLPaaajugqbiabg2da 9OWaaeWaaeaadaaeqbqaaiaadwfadaWgaaWcbaGaamyAaaqabaGcca WGybWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgaaeqaniabggHiLdaa kiaawIcacaGLPaaadaqadaqaamaaqafabaGaamOvamaaBaaaleaaca WGQbaabeaakiaadMfadaWgaaWcbaGaamOAaaqabaaabaGaamOAaaqa b0GaeyyeIuoaaOGaayjkaiaawMcaaiabg2da9maaqafabaGaamyvam aaBaaaleaacaWGPbaabeaaaeaacaWGPbGaamOAaaqab0GaeyyeIuoa kiaadAfadaWgaaWcbaGaamOAaaqabaGccaWGybWaaSbaaSqaaiaadM gaaeqaaOGaamywamaaBaaaleaacaWGQbaabeaaaaa@5AC2@ in terms of the coordinates of the vectors. The dyadic product is then the tensor with components T ij = U i V j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadsfadaWgaaWcba qcLboacaWGPbGaamOAaaWcbeaakiabg2da9KqzafGaamyvaOWaaSba aSqaaKqzGdGaamyAaaWcbeaajugqbiaadAfakmaaBaaaleaajug4ai aadQgaaSqabaaaaa@4626@ .

Note 2 to entry: The dyadic product of two vectors is denoted by UV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxkcXca WHwbaaaa@3C93@ or UV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacaWHwbaaaa@3A8A@ .


fr
produit tensoriel, <de deux vecteurs> m
pour deux vecteurs U et V d'un espace euclidien à n dimensions, tenseur du deuxième ordre défini par la forme bilinéaire f(X,Y)=(UX)(VY) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadAgacaGGOaGaaC iwaiaabYcacaaMe8UaaCywaiaacMcacqGH9aqpcaGGOaGaaCyvaiab gwSixlaahIfacaGGPaGaaiikaiaahAfacqGHflY1caWHzbGaaiykaa aa@4ADC@ , où X et Y sont des vecteurs quelconques du même espace

Note 1 à l'article: 1 La forme bilinéaire peut être représentée par f( X,Y )=( i U i X i )( j V j Y j )= ij U i V j X i Y j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaKqzafGaamOzaOWaae WaaeaacaWHybGaaiilaiaahMfaaiaawIcacaGLPaaajugqbiabg2da 9OWaaeWaaeaadaaeqbqaaiaadwfadaWgaaWcbaGaamyAaaqabaGcca WGybWaaSbaaSqaaiaadMgaaeqaaaqaaiaadMgaaeqaniabggHiLdaa kiaawIcacaGLPaaadaqadaqaamaaqafabaGaamOvamaaBaaaleaaca WGQbaabeaakiaadMfadaWgaaWcbaGaamOAaaqabaaabaGaamOAaaqa b0GaeyyeIuoaaOGaayjkaiaawMcaaiabg2da9maaqafabaGaamyvam aaBaaaleaacaWGPbaabeaaaeaacaWGPbGaamOAaaqab0GaeyyeIuoa kiaadAfadaWgaaWcbaGaamOAaaqabaGccaWGybWaaSbaaSqaaiaadM gaaeqaaOGaamywamaaBaaaleaacaWGQbaabeaaaaa@5AC2@ en fonction des coordonnées des vecteurs. Le produit tensoriel est donc le tenseur de coordonnées T ij = U i V j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadsfadaWgaaWcba qcLboacaWGPbGaamOAaaWcbeaakiabg2da9KqzafGaamyvaOWaaSba aSqaaKqzGdGaamyAaaWcbeaajugqbiaadAfakmaaBaaaleaajug4ai aadQgaaSqabaaaaa@4626@ .

Note 2 à l'article: Le produit tensoriel est noté UV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacqGHxkcXca WHwbaaaa@3C93@ ou UV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahwfacaWHwbaaaa@3A8A@ .


de
Tensorprodukt (zweier Vektoren), n
dyadisches Produkt, n

es
producto tensorial (de dos vectores)
producto diádico

ja
二項積
テンソル積, <2つのベクトルの>

pl
iloczyn tensorowy (dwóch wektorów)

pt
produto tensorial

sr
тензорски производ, <два вектора> м јд

sv
dyadisk produkt
tensorprodukt (av två vektorer)

zh
并向量积
张量积, <两个向量的>

Publication date: 2008-08
Copyright © IEC 2017. All Rights Reserved.