IEC 60050 - International Electrotechnical Vocabulary


IEVref:	102-03-47		ID:
Language:	en		Status: Obsolete

Term:	inner product (of a tensor and a vector)
Synonym1:	contracted product (of a tensor and a vector) [Preferred]
Synonym2:
Synonym3:
Symbol:
Definition:	for a tensor of the second order $T = (T_{i j})$ and a vector U = (U_k), vector the components of which are given by ${(T \cdot U)}_{i} = \sum_{m} T_{i m} U_{m}$ NOTE 1 The inner product of two vectors is their scalar product, because a tensor of the first order is considered as a vector. NOTE 2 An example is the relation $D = \vec{\vec{ε}} \cdot E$ between the electric field strength E and the electric flux density D, where $\vec{\vec{ε}}$ is the absolute permittivity of an anisotropic medium. NOTE 3 The inner product of a tensor and a vector is denoted by a half-high dot (·) between the two symbols.
Publication date:	2007-08
Source
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Internal notes:	2017-06-02: Cleanup - Remove Attached Image 102-03-47en.gif
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T=( T ij ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaGqadKqzGfGaa8hvaO Gaeyypa0JaaiikaiaadsfadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa aiykaaaa@3C54@ UU_k (T⋅U) 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 The inner product of two vectors is their scalar product, because a tensor of the first order is considered as a vector. NOTE 2 An example is the relation

D = \vec{\vec{ε}} \cdot E

between the electric field strength E and the electric flux density D, where

\vec{\vec{ε}}

is the absolute permittivity of an anisotropic medium. NOTE 3 The inner product of a tensor and a vector is denoted by a half-high dot (·) between the two symbols.