Use of tensors for dielectric permittivity and magnetic permeability

  • Thread starter EmilyRuck
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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!

Emily
 

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  • #2
tiny-tim
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Hello Emily! :smile:
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?
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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