IEVref: 102-03-39 ID: Language: en Status: Standard Term: tensor of the second order Synonym1: tensor Synonym2: Synonym3: Symbol: 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 U and V the scalar $\sum _{i,j=1}^{n}{T}_{ij}{U}_{i}{V}_{j}$, where ${U}_{i}$ and ${V}_{j}$ are the coordinates of vectors U and V. 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 $\stackrel{\to }{\stackrel{\to }{T}}$. The tensor $T$ with components ${T}_{ij}$ can be denoted $\left({T}_{ij}\right)$. 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 Source: Replaces: Internal notes: 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: