Use of tensors for dielectric permittivity and magnetic permeability

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

    [itex]\mathbf{D} = \mathbf{\epsilon} \cdot \mathbf{E}\\
    \mathbf{B} = \mathbf{\mu} \cdot \mathbf{H}[/itex]

    But in a generic medium (non linear, non isotropic, non homogeneous) [itex]\mathbf{\epsilon}[/itex] and [itex]\mathbf{\mu}[/itex] are tensors. Now, why not matrices with dimension 3x3? [itex]\mathbf{E}[/itex] and [itex]\mathbf{H}[/itex] 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 [itex]\mathbf{\epsilon}[/itex] and [itex]\mathbf{\mu}[/itex] tensors and not just matrices?
    Thank you anyway!

  2. jcsd
  3. tiny-tim

    tiny-tim 26,041
    Science Advisor
    Homework Helper

    Hello Emily! :smile:
    Yes, a tensor is an operator with an input and an output …

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

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