View source for 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.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. Dubrovin,   A.T. Fomenko,   S.P. Novikov,   "Modern geometry - methods and applications" , Springer  (1984)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W.H. Greub,   "Multilinear algebra" , Springer  (1967)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  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}}</TD></TR>
</table>

====Comments====
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 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====
<table>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  H. Jeffreys ''Cartesian tensors'' (7th imp.) Cambridge University Press [1931] (1969) ISBN 0-521-09191-8 {{ZBL|57.0974.01}}</TD></TR>
</table>

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