Affine tensor

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