Lots of physical quantities of interest can be described by tensors, and a small subset of the common ones is shown in the flash animation below. The tensors in the circles are those that can be applied and measured in any orientation with respect to the crystal (e.g. stress, electric field) and are known as field tensors. The tensors that link these properties are those that are intrinsic properties of the crystal and must conform to its symmetry (e.g. thermal conductivity), and are known as matter tensors.
Many of these quantities are described by symmetrical tensors (e.g. stress, electrical susceptibility), for which the off diagonal components Tij and Tji are equal (i.e. T12 = T21). Taking electrical susceptibility as an example, this means that applying a field in the 1 direction, produces a polarisation in the 2 direction, and this is equal in magnitude to the polarisation produced in the 1 direction if the field is applied in the 2 direction. Whilst this seems intuitively reasonable, the explanation for it is not immediately obvious and the mathematical proof is in fact quite complex. Readers who would like to follow the detailed argument can refer to the textbook by Nye (Physical Properties of Crystals: see the Going further page)