| bilinear form defined for any pair of vectors of an n-dimensional Euclidean vector space
Note 1 to entry: For a given orthonormal base, a tensor of the second order can be represented by components , generally presented in the form of a square matrix, such that attributes to the pair of vectors U and V the scalar , where and 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 ou . The tensor with components can be denoted .
Note 4 to entry: A complex tensor is defined by a real part and an imaginary part: where and are real tensors.