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## Determination of electromagnetic medium from the Fresnel Surface

#### M.F. Dahl

• preprint at arxiv.
• updated version. Note. Since the updated version contains corrections in response to reviewer comments, it has not been placed on the arxiv.
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 Vξ: [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(adjκ) + (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 Gijkl0 ξi ξjξkξl = Gijkl ξi ξjξkξl [ 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.