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IEVref: | 102-03-47 | ID: | |

Language: | en | Status: Standard | |

Term: | inner product, <of a tensor and a vector> | ||

Synonym1: | contracted product, <of a tensor and a vector> [Preferred] | ||

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Symbol: | |||

Definition: | for a tensor of the second order $T=({T}_{ij})$ and a vector = (UU), vector the components of which are given by ${(T\cdot U)}_{i}={\displaystyle \sum _{m}{T}_{im}}{U}_{m}$_{k}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 $D=\overrightarrow{\overrightarrow{\epsilon}}\cdot E$ between the electric field strength , where $\overrightarrow{\overrightarrow{\epsilon}}$ is the absolute permittivity of an anisotropic medium. DNote 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 | ||

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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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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 $D=\overrightarrow{\overrightarrow{\epsilon}}\cdot E$ between the electric field strength ** E** and the electric flux density

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