As far as I can tell, technically MKPolyline and MKPolygon should not be functional/available when running on an iPad or any other device prior to iOS4.  Nevertheless, when I installed 3.2.2, on the device, and updated my build tools with the latest (non-beta) iOS 4.2 SDK, they both work just fine, along with their associated views.  What the hell?

I'm not exactly complaining here, given that the alternative would be NVPolyline and it doesn't play well with a huge number of points.  I'm just confused.  Can someone explain why these classes are available and working on the iPad?

I have not, however, had any success subclassing MKOverlayPathView on 3.2.2, even though MKPolylineView and MKPolygonView both descend from it.


Question: Can Gauss' and Ampere's Laws be written in terms of the divergence of an energy four-vector?
Answer: The covariant formulation of EM is precisely this. The formulation as a gauge theory also does this. ($c = 1$ in the following)

Given the $E$- and $B$-fields as spatial three-vectors in some frame, we construct the antisymmetric field strength tensor as (roman indices are spatial indices, summation over repeated indices implied)

$$ F^{0i} := E^i \; \text{and} \; F^{ij} := \epsilon^{ijk}B^k$$

Using the behaviour of the fields under Lorentz transformations, one can indeed see that this $F_{\mu\nu}$ is a proper tensor on Minkowski space. Similarly, we construct a four-current out of the classical current $\vec j$  and the charge density $\rho$ by

$$ j^i = j^i \; \text{and} \; j^0 = \rho$$

Maxwell's laws now simply read

$$ \partial_\mu F^{\mu\nu} = \mu_0j^\nu \; \text{and} \; \partial_\mu \epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma} = 0$$



This is not yet the most abstract way to present this. Using the language of differential forms, we have forms $F = F_{\mu\nu}\mathrm{d}x^\mu \wedge \mathrm{d}x^\nu$ and $j = j_\mu \mathrm{d}x^\mu$. Maxwell's laws now look, in their most concise form, like this:

$$ \mathrm{d}F = 0 \; \text{and} \; \mathrm{d}\star F = \star j$$

where the star is the Hodge dual. One advantage of this language is that one may conclude that there is a 1-form $A$ with $F = \mathrm{d}A$ on any contractible subset of our space. More precisely, it exists if the second deRham cohomology vanishes.

The gauge theory description of electrodynamics starts from gluing together these $A$ which always exist locally to get a globally defined gauge potential $A$.