Namespaces
Variants
Actions

Affine tensor

From Encyclopedia of Mathematics
Jump to: navigation, search
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 $p$ copies of an $n$-dimensional vector space $E$ and $q$ copies of the dual vector space $E^*$. Such a tensor is said to be of type $(p,q)$, the number $p+q$ defining the valency, or degree, of the tensor. Having chosen a basis $\{e_i\}$ in $E$, one defines an affine tensor of type $(p,q)$ with the aid of $n^{p+q}$ components $T^{i_1\ldots i_p}_{j_1\ldots j_p}$ which transform as a result of a change of basis $e'_i = A_i^s e_s$ according to the formula $$ T'^{i_1\ldots i_p}_{j_1\ldots j_p} = A'^{i_1}_{s_1} \cdots A'^{i_p}_{s_p} A^{t_1}_{j_1} \cdots A^{t_q}_{j_q} T^{i_1\ldots i_p}_{j_1\ldots j_p} $$ where $A^s_j A'^i_s = \delta^i_j$. 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.

The tensor $\delta^i_j$ is the Kronecker delta tensor.

An isotropic tensor is one for which the components are unchanged under change of basis. The Kronecker delta tensor is isotropic; in dimension $n=3$ the discriminant tensor $\epsilon_{ijk}$ defined by $\epsilon_{123} = \epsilon_{231} = \epsilon_{312} = 1$, $\epsilon_{321} = \epsilon_{213} = \epsilon_{132} = -1$, all other values zero, of order 3, is isotropic.

See also: Contravariant tensor, Covariant 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) Zbl 0369.53012 Graduate Texts in Mathematics 130 (2nd ed.) Springer (1991) ISBN 3-540-52018-X Zbl 0732.53002
[b1] H. Jeffreys Cartesian tensors (7th imp.) Cambridge University Press [1931] (1969) ISBN 0-521-09191-8 Zbl 57.0974.01
How to Cite This Entry:
Affine tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_tensor&oldid=54534
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article