# Affine tensor

From Encyclopedia of Mathematics

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.

#### Comments

An affine tensor as described above is commonly called simply a tensor.

#### References

[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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Affine_tensor&oldid=17159

This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article