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IEVref: | 102-03-39 | ID: | |

Language: | en | Status: Standard | |

Term: | tensor of the second order | ||

Synonym1: | tensor | ||

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Definition: | bilinear form defined for any pair of vectors of an n-dimensional Euclidean vector spaceNote 1 to entry: For a given orthonormal base, a tensor $T$ of the second order can be represented by ${n}^{2}$ components ${T}_{ij}$, generally presented in the form of a square matrix, such that $T$ attributes to the pair of vectors the scalar $\sum _{i,j=1}^{n}{T}_{ij}}{U}_{i}{V}_{j$, where ${U}_{i}$ and ${V}_{j}$ are the coordinates of vectors V and U. VNote 2 to entry: A tensor of the second order can be defined by a bilinear form applied to two vectors (covariant tensor), to two linear forms (contravariant tensor), or to a vector and a linear form (mixed tensor). This distinction is not necessary for a Euclidean space. It is also possible to generalize to tensors of order Note 3 to entry: A tensor is indicated by a letter symbol in bold-face sans-serif type or by two arrows above a letter symbol: Note 4 to entry: A complex tensor $T$ is defined by a real part and an imaginary part: $T=A+jB$ where $A$ and $B$ are real tensors. | ||

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 | ||

CO remarks: | |||

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VT remarks: | |||

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Note 1 to entry: For a given orthonormal base, a tensor $T$ of the second order can be represented by ${n}^{2}$ components ${T}_{ij}$, generally presented in the form of a square matrix, such that $T$ attributes to the pair of vectors ** U** and

Note 2 to entry: A tensor of the second order can be defined by a bilinear form applied to two vectors (covariant tensor), to two linear forms (contravariant tensor), or to a vector and a linear form (mixed tensor). This distinction is not necessary for a Euclidean space. It is also possible to generalize to tensors of order *n* defined by *n*-linear forms and for which the components have *n* indices. Tensors of order 1 are considered as vectors and tensors of order 0 are considered as scalars.

Note 3 to entry: A tensor is indicated by a letter symbol in bold-face sans-serif type or by two arrows above a letter symbol: ** T** or $T$ ou $\overrightarrow{\overrightarrow{T}}$. The tensor $T$ with components ${T}_{ij}$ can be denoted $({T}_{ij})$.

Note 4 to entry: A complex tensor $T$ is defined by a real part and an imaginary part: $T=A+jB$ where $A$ and $B$ are real tensors.