Affine tensor
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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
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