# Use of tensors for dielectric permittivity and magnetic permeability

1. Feb 28, 2013

### EmilyRuck

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

$\mathbf{D} = \mathbf{\epsilon} \cdot \mathbf{E}\\ \mathbf{B} = \mathbf{\mu} \cdot \mathbf{H}$

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

2. Feb 28, 2013

### tiny-tim

Hello Emily!
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.