An element of the tensor product of copies of an -dimensional vector space and copies of the dual vector space . Such a tensor is said to be of type , the number defining the valency, or degree, of the tensor. Having chosen a basis in , one defines an affine tensor of type with the aid of components which transform as a result of a change of basis according to the formula
where . It is usually said that the tensor components undergo a contravariant transformation with respect to the upper indices, and a covariant transformation with respect to the lower.
An affine tensor as described above is commonly called simply a tensor.
|[a1]||B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Modern geometry - methods and applications" , Springer (1984) (Translated from Russian)|
|[a2]||W.H. Greub, "Multilinear algebra" , Springer (1967)|
|[a3]||C.T.J. Dodson, T. Poston, "Tensor geometry" , Pitman (1977)|
Affine tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_tensor&oldid=17159