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##
Determination of electromagnetic medium from the Fresnel Surface

#### M.F. Dahl

The paper relies on some computations that require computer algebra. Below are
Mathematica toolbooks for these computations.
### Notebooks for paper

- Proposition 2.2: Computer algebra computations for proof of Proposition 2.2:
[notebook,
pdf]
- Proposition 3.3: A computer verification of the polarisation identity for (4,0)-tensors on a
*4*-manifold:
[notebook,
pdf]
- Section 3.2: Isotropic medium:
[
notebook,
pdf]
- Proposition 3.4: Compute Frensel equation for Hodge operator:
[notebook,
pdf
].
- Proposition 3.5: Compute spectrum for
*4x4* matrix *H*:
[notebook,
pdf]
- Example 3.6: Plot for Figure 1:
[notebook,
pdf]
- Example 3.6: Compute dimensions of
*V*_{ξ}:
[notebook,
pdf]
- Theorem 4.2: Computer algebra computations for implication
* (i) ⇒ (ii)*:
[
notebook,
pdf]
- Lemma 4.3: Diagonal elements in
*A* can be ordered in arbitrary way:
[notebook,
pdf]
- Lemma 4.3: Case B is not possible:
[notebook,
pdf]
- Lemma 4.3: Computations for Case C:
[
notebook,
pdf]
- Lemma 4.6: For the computer algebra computations for this lemma,
see homepage
of
*Non-dissiative electromagnetic media with two Lorentz nullcones*:
[link]
- Theorem 5.1
*(iv)*: Script for proving *G(*adj*κ) + (det κ)^2 G(κ)=0*:
[input script,
output]
- Example 5.2:
*5*-parameter medium with vanishing Tamm-Rubilar tensor density. This shows that the
first arrow in diagram (1) is not injective:
[notebook,
pdf]
- Example 5.3: Complex medium with the same Fresnel surface as the Minkowski metric:
[notebook,
pdf]
- Example 5.4: Latter arrow in diagram (52) is not injective:
[notebook,
pdf]

Please note: The numbering in the above notebooks follow the paper
submitted for review (see above). This differs slightly from the
preprint on arxiv.
### Supplementary notebooks

- Various identities for
*3x3* and *4x4* matrices and the Levi-Civita permutation symbol:
[notebook,
pdf]
- Section 3.1: Verify that
*
G*^{ijkl}_{0}
ξ_{i}
ξ_{j}ξ_{k}ξ_{l}
=
G^{ijkl}
ξ_{i}
ξ_{j}ξ_{k}ξ_{l}
[
notebook,
pdf]
- Proposition 3.1: Alternative computer algebra verification of implication
*(ii) ⇒ (i)*:
[notebook,
pdf]
- Verify that the definition of the Tamm-Rubilar tensor density coincides with the definition in Hehl-Obukhov:
[notebook,
pdf]
- Example of computation of Groebner basis in Mathematica.
[notebook,
pdf]

### kappaLib: A library for manipulation of electromagnetic
medium

The above notebooks make use of
KappaLib version 1.1, which is a
collection of Mathematica routines for manipulating electromagnetic
medium tensors. To install this library, first download the file onto
your harddrive. Then load kappaLib into a Mathematica session by
typing
<<kappaLib.m

If this generates an error, Mathematica can not find the file. The
command Directory[] shows what directory Mathematica is currently in,
and you can use SetDirectory["/user/mydir"] to change this directory.

Last modified 12.7.2012.