In an isotropic material, some physical responses are the same in every direction. In an anisotropic material, the response depends on orientation, so one scalar material coefficient may no longer be enough.

Conductivity tensor

For isotropic electrical conduction,

$$\mathbf J=\sigma\mathbf E,$$

where scalar conductivity $\sigma$ makes current density parallel to the electric field.

In an anisotropic conductor, conductivity becomes a second-order tensor:

$$\mathbf J=\boldsymbol\sigma\mathbf E.$$

The current density need not point in the same direction as the electric field.

Principal material directions

For a symmetric conductivity tensor, there is an orthogonal basis in which the tensor is diagonal. Along those principal directions, each electric-field component produces current only along the same direction.

Rotating to another coordinate system introduces off-diagonal components even though the material itself has not changed.

Stress as a tensor

Stress provides another second-order tensor. A surface with unit normal $\mathbf n$ experiences a traction vector

$$\mathbf t=\boldsymbol\sigma\mathbf n.$$

The stress matrix used in plane-stress calculations is therefore a coordinate representation of a tensorial physical state.