IEVref:102-03-47ID:
Language:enStatus: Standard
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 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 (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 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.


Publication date:2008-08
Source
Replaces:
Internal notes:2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO
2017-08-24: Corrected order of </i> and </sub> tags. LMO
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VT remarks:
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