Use of tensors for dielectric permittivity and magnetic permeability

[EmilyRuck]

Hello!
In the study of electric and magnetic fields, two equations are called the constitutive relations of the medium (the vacuum, for example):

\mathbf{D} = \mathbf{\epsilon} \cdot \mathbf{E}\\<br /> \mathbf{B} = \mathbf{\mu} \cdot \mathbf{H}

But in a generic medium (non linear, non isotropic, non homogeneous) \mathbf{\epsilon} and \mathbf{\mu} are tensors. Now, why not matrices with dimension 3x3? \mathbf{E} and \mathbf{H} are "simple" three-dimensional vectors. I know that a matrix is a particular case of a tensor, but so why do we never use the term "matrix" in this context?
A matrix could exist only if a particolar system of coordinates is defined, whereas a tensor can always exist: is it the reason for calling \mathbf{\epsilon} and \mathbf{\mu} tensors and not just matrices?
Thank you anyway!

Emily

 

[tiny-tim]

Hello Emily!

A matrix could exist only if a particolar system of coordinates is defined, whereas a tensor can always exist: is it the reason for calling \mathbf{\epsilon} and \mathbf{\mu} tensors and not just matrices?

Yes, a tensor is an operator with an input and an output …

you put one vector in, another vector (not necessarily parallel) comes out!

You don't need the coordinates (though of course they often help a lot), any more than you need coordinates to write a vector.