Affine tensor

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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)
How to Cite This Entry:
Affine tensor. A.P. Shirokov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098