Electromagnetic fields from contact- and symplectic geometry

M.F. Dahl

preprint

Mathematica toolbooks

The paper relies on some computations that are best carried out with computer algebra. Below are Mathematica toolbooks for these computations.

Section 2.1: identity tensor has trace 6. [.nb, .pdf ]
Section 2.1: scalar product on (2,2)-tensors has signature (16,21) [.nb, .pdf ]
Proof of Theorem 2.1: the two expressions for Z coincide [.nb, .pdf ]
Proof of Theorem 2.1: Z is non-degenerate [.nb, .pdf ]
Proof of Theorem 2.1: proof of identity for the Levi-Civita permutation symbol [.nb, .pdf ]
Proof of Theorem 2.1: W and Z are orthogonal. [.nb, .pdf ]
Proof of Theorem 2.1: (U+Z)^⊥ ⊂ W [.nb, .pdf ]
Proof of Proposition 2.2: ⁽²⁾kappa ∈ U^⊥ [ .nb , .pdf ]
Figures 1 and 2: Plots of contact structures [.nb, .pdf ]
Example 3.6: Contact forms in Examples 3.1 and 3.6 are globally diffeomorphic [.nb, .pdf ]
Section 3.3: Decomposition of TE₁₁ and TM₁₁ solutions into Beltrami fields and finding sets where the contact condition fails [.nb, .pdf ]
Figures 3 and 4: Plots of contact structures from TM₁₁ and TE₁₁ fields [.nb, .pdf ]
Figure 5: Sets where contact condition fails for TE₁₁ and TM₁₁ solutions [.nb, .pdf ]
Example 3.10: compatible contact forms [.nb, .pdf ]
Section 3.5: The value of a vector field and its curl at one point are arbitrary. This result is implicitly used when calculating how many degrees of freedom the medium in Theorem 3.13 depends upon [.nb, .pdf ]
Example 3.16: Calculation of explicit expression for permittivity matrix [.nb, .pdf ]
Example 3.16: Double check that solution solves Maxwell's equations [.nb, .pdf ]
Section 4: Proof that the map in Proposition 4.1 coincide with the usual star map on a symplectic manifold [.nb, .pdf ]
Proof of Proposition 4.3: Computations related to proof of invertibility of kappa [.nb, .pdf ]

Last modified 4 June 2010