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-   -   Use of tensors for dielectric permittivity and magnetic permeability (http://www.physicsforums.com/showthread.php?t=675163)

 EmilyRuck Feb28-13 10:35 AM

Use of tensors for dielectric permittivity and magnetic permeability

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

 tiny-tim Feb28-13 05:59 PM

Hello Emily! :smile:
Quote:
 Quote by EmilyRuck (Post 4289441) 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! :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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