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

IEV ref102-03-47

en
inner product, <of a tensor and a vector>
contracted product, <of a tensor and a vector>
for a tensor of the second order T=( T ij ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaGqadKqzGfGaa8hvaO Gaeyypa0JaaiikaiaadsfadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa aiykaaaa@3C54@ and a vector U = (Uk), vector the components of which are given by (TU) i = m T im U m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcaieWajugybi aa=rfakiabgwSixlaahwfacaGGPaWaaSbaaSqaaiaadMgaaeqaaOGa eyypa0ZaaabuaeaacaWGubWaaSbaaSqaaiaadMgacaWGTbaabeaaae aacaWGTbaabeqdcqGHris5aOGaamyvamaaBaaaleaacaWGTbaabeaa aaa@4923@

Note 1 to entry: The inner product of two vectors is their scalar product, because a tensor of the first order is considered as a vector.

Note 2 to entry: An example is the relation D= ε E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahseacqGH9aqpda WhkaqaamaaFOaabaGaeqyTdugacaGLrdcaaiaawgniaiabgwSixlaa hweaaaa@3F4D@ between the electric field strength E and the electric flux density D, where ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaayIW7daWhkaqaam aaFOaabaGaeqyTdugacaGLrdcaaiaawgniaaaa@3BF3@ is the absolute permittivity of an anisotropic medium.

Note 3 to entry: The inner product of a tensor and a vector is denoted by a half-high dot (·) between the two symbols.


fr
produit intérieur, <d'un tenseur et d'un vecteur> m
produit contracté, <d'un tenseur et d'un vecteur> m
pour un tenseur du deuxième ordre T=( T ij ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaGqadKqzGfGaa8hvaO Gaeyypa0JaaiikaiaadsfadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa aiykaaaa@3C54@ et un vecteur U = (Uk), vecteur dont les coordonnées sont (TU) i = m T im U m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcaieWajugybi aa=rfakiabgwSixlaahwfacaGGPaWaaSbaaSqaaiaadMgaaeqaaOGa eyypa0ZaaabuaeaacaWGubWaaSbaaSqaaiaadMgacaWGTbaabeaaae aacaWGTbaabeqdcqGHris5aOGaamyvamaaBaaaleaacaWGTbaabeaa aaa@4923@

Note 1 à l'article: Le produit intérieur de deux vecteurs est leur produit scalaire puisqu'un tenseur du premier ordre est considéré comme un vecteur.

Note 2 à l'article: Un exemple est la relation D= ε E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahseacqGH9aqpda WhkaqaamaaFOaabaGaeqyTdugacaGLrdcaaiaawgniaiabgwSixlaa hweaaaa@3F4D@ entre le champ électrique E et l'induction électrique D, où ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaayIW7daWhkaqaam aaFOaabaGaeqyTdugacaGLrdcaaiaawgniaaaa@3BF3@ est la permittivité absolue d'un milieu anisotrope.

Note 3 à l'article: Le produit intérieur d'un tenseur et d'un vecteur est indiqué par un point à mi-hauteur (·) entre les deux symboles.


de
inneres Produkt (eines Tensors mit einem Vektor), n

es
producto interior (de un tensor y un vector)

ja
内積, <テンソルとベクトルとの>
縮約積, <テンソルとベクトルとの>

pl
iloczyn wewnętrzny (tensora i wektora)
iloczyn przez kontrakcję (tensora i wektora)

pt
produto interior (de um tensor e de um vector)
produto contratado (de um tensor e de um vector)

sr
унутрашњи производ, <тензора и вектора> м јд
скраћени производ, <тензора и вектора> м јд

sv
inre produkt (av en tensor och en vektor)

zh
内积, <一个张量和一个向量的>
收缩积, <一个张量和一个向量的>

Publication date: 2008-08
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