What is a Tensor?
Tensors are simply mathematical objects that can be used to describe physical properties, just like scalars and vectors. In fact tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor, and a vector is a first rank tensor.
The rank (or order) of a tensor is defined by the number of directions (and hence the dimensionality of the array) required to describe it. For example, properties that require one direction (first rank) can be fully described by a 3×1 column vector, and properties that require two directions (second rank tensors), can be described by 9 numbers, as a 3×3 matrix. As such, in general an n^{th} rank tensor can be described by 3^{n} coefficients.
The need for second rank tensors comes when we need to consider more than one direction to describe one of these physical properties. A good example of this is if we need to describe the electrical conductivity of a general, anisotropic crystal. We know that in general for isotropic conductors that obey Ohm's law:
j = σEWhich means that the current density j is parallel to the applied electric field, E and that each component of j is linearly proportional to each component of E. (e.g. j_{1} = σE_{1}).
However in an anisotropic material, the current density induced will not necessarily be parallel to the applied electric field due to preferred directions of current flow within the crystal (a good example of this is in graphite). This means that in general each component of the current density vector can depend on all the components of the electric field:
j_{1} = σ_{11}E_{1} + σ_{12}E_{2} + σ_{13}E_{3}j_{2} = σ_{21}E_{1} + σ_{22}E_{2} + σ_{23}E_{3}
j_{3} = σ_{31}E_{1} + σ_{32}E_{2} + σ_{33}E_{3}
So in general, electrical conductivity is a second rank tensor and can be specified by 9 independent coefficients, which can be represented in a 3×3 matrix as shown below:
σ = 

Other examples of second rank tensors include electric susceptibility, thermal conductivity, stress and strain. They typically relate a vector to another vector, or another second rank tensor to a scalar. Tensors of higher rank are required to fully describe properties that relate two second rank tensors (e.g. Stiffness (4th rank): stress and strain) or a second rank tensor and a vector (e.g. Piezoelectricity (3rd rank): stress and polarisation).
To view these and more examples, and to investigate how changing the components of the tensors affect these properties, go through the flash program below.