# Tensors

Tensor analysis can be used to describe this situation. Consider an anisotropic material with principal axes *1*,
*2* and *3*. For sake of example, a temperature gradient, *gradT*, is applied in an arbitrary direction
perpendicular to the *3*-axis, non-parallel to the principal axes *1* and *2*. The resulting heat flow,
** J**, will not be parallel to the temperature gradient. The components of these vectors parallel
to the principal axes are as shown in the diagram below.

As mentioned before, J = k(*gradT*). Using tensor notation
to describe the above situation,

J_{1} = k_{11}(*gradT*_{1}),
J_{2} = k_{22}(*gradT*_{2}), and
J_{3} = 0

In general,

J_{i} = k_{ij}(*gradT _{j}*)

where the suffices i and j represent the relevant axes. This equation summarises the following matrix equation:

Here J_{i} and *gradT _{j}* are first rank tensors (one suffix) and k

_{ij}is second rank (two suffices). Since

*1*,

*2*and

*3*are the principal axes then the

*k*tensor is a diagonal matrix - all values are zero unless i = j. If an arbitrary set of axes is chosen, then it will be a general matrix.

_{ij}Other second rank tensors, which relate first rank tensors, include:

- Electrical conductivity, relating current density to electric field
- Permittivity, relating dielectric displacement to electric field
- Permeability, relating magnetic induction to magnetic field

Thermal expansion is also a second rank tensor, but it relates strain (a second rank tensor) to temperature change (a scalar - zero rank).