A little clarification on Cartesian tensor notation

[Kashmir]

Goldstein pg 192, 2 ed


In a Cartesian three-dimensional space, a tensor $\mathrm{T}$ of the $N$ th rank may be defined for our purposes as a quantity having $3^{N}$ components $T_{i j k}$.. (with $N$ indices) that transform under an orthogonal transformation of coordinates, $\mathbf{A}$) according to the following scheme:*
$T_{i j k \ldots}^{\prime} \left(\mathbf{x}^{\prime}\right)=a_{i l} a_{j m} a_{k n} \ldots T_{l m n \ldots}(\mathbf{x})$

I've just a small doubt here:

What do the $\mathbf{x}$ mean here, are they vectors that multiply $T$ or is the author using them to signify the coordinate system in which the components of $T$ are calculated?

 

[ergospherical]

In general the tensor is a function of the position $\mathbf{x} \in \mathbf{R}^3$ in space (e.g. stress, strain, electrical conductivity, etc. all depend on the position within the material). After changing coordinates, every point $p$ has been re-labelled by a new set of position coordinates $\mathbf{x} \rightarrow \mathbf{x}' = \mathbf{A} \mathbf{x}$; hence why you also put a prime on the argument.

 

[Kashmir]

In general the tensor is a function of the position $\mathbf{x} \in \mathbf{R}^3$ in space (e.g. stress, strain, electrical conductivity, etc. all depend on the position within the material). After changing coordinates, every point $p$ has been re-labelled by a new set of position coordinates $\mathbf{x} \rightarrow \mathbf{x}' = \mathbf{A} \mathbf{x}$; hence why you also put a prime on the argument.

Thank you. Makes it perfectly clear! I :)