Exterior Form: Fill & Download for Free

Warning: Some math is involved.The modern starting point is to postulate the existence of a 4-dimensional vector field with a massless current. If the vector field is (at least) three times differentiable and if the spacetime in which it exists is endowed with a metric, Maxwell’s equations follow as mathematical identities.To get down into the nitty-gritty of things: Let the vector field be denoted [math]A_\mu. [/math]The Maxwell tensor is defined as [math]F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu.[/math] Using the language of exterior forms, this can be written as[math]F={\rm d}A.\tag*{}[/math]Here comes the first demonstration of the power of exterior forms: the exterior derivative is nilpotent, meaning [math]{\rm d}^2=0.[/math] therefore,[math]{\rm d}F={\rm d}^2A=0.\tag*{}[/math]Spelled out in component form (which I am not going to do here; it’s not hard, but it is tedious*), this is two of Maxwell’s famed equations: Faraday’s law and Gauss’s law for magnetism.Exterior forms have duals. These are formed using the Levi-Civita symbol and the metric (hence the need for a metric). They are usually denoted by a star.Using the dual and the exterior derivative, we can define a massless current:[math]J={\star{\rm d}}{\star F}.\tag*{}[/math]This definition amounts to Gauss’s law and Ampère’s law, i.e., the other pair of Maxwell’s equations.The current is conserved. This follows from the fact that the dual is its own inverse and the exterior derivative is nilpotent:[math]{\star{\rm d}}{\star J}={\star{\rm d}\star}{\star{\rm d}{\star F}}={\star{\rm d}}^2{\star F}=0.\tag*{}[/math]This is the equation of charge and current conservation.All these results can be spelled out in component form and, ultimately, in the usual three-dimensional form. The icing on the cake is that they remain valid even in the curved spacetime of general relativity.I find this approach immensely powerful: to be able to derive pretty much everything we know about electromagnetism in just a few short lines of equations tells us just how powerful these conceptual tools of mathematics really are.Of course just because we can derive these equations does not mean that they correctly describe Nature. Is a three times differentiable vector field with a conserved current a valid description of electromagnetic phenomena? This can only be determined by experiment. And if experiment said otherwise, we would have to modify the theory. For instance, we might have to make the field massive by defining the current differently: [math]J={\star{\rm d}}{\star F}+\mu^2A.[/math] This yields the so-called Proca theory, named after the Romanian physicist Alexandru Proca.Lastly, I should mention that there is another way to derive Maxwell’s equations, from the Lagrangian [math]{\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-A_\mu J^\mu.[/math] This seems superfluous in light of the fact that the field equations can be derived, as I showed above, from the existence of [math]A_\mu[/math] alone. However, the Lagrangian formalism has the advantage that it is easy to introduce other fields and forces. It also directly leads to a Hamiltonian formalism that, in turn, can be used to derive the quantum version of electromagnetism.*I am presenting some additional details on my Web site: A covariant form of Maxwell's equationsI thank my generous supporters on Patreon. If you like my answers, please consider joining them.

We’ve seen it in a couple episodes; Hell Bent, Name of the Doctor etc“What kind of idiot, would steal a faulty TARDIS?”

There are three commonly used methods. The two similar methods are cut rifling and button rifling. In cut rifling the bore is first finished to the proper diameter. The rifling tool, which has tiny chisel-like cutters arranged on it, is pushed through the bore repeatedly, cutting the grooves a little deeper at each pass. The rifling machine imparts the desired spiral, or twist rate, to the tool as it’s cutting. This method has been used for hundreds of years, and is well proven. However, it is slow, and the tiny cutters are prone to failure. If a cutter breaks during the rifling process, it usually means a ruined barrel. Cut rifling also leaves tiny burrs on the edges of the grooves, which means the barrel must be broken in before it achieves its best performance.Button rifling is very similar, but instead of little cutters, the rifling head has little buttons that are forced into the steel of the barrel as it’s pushed through the bore. This method is also slow, but less prone to failure in process, and leaves much smoother grooves. Bear in mind that both of these methods only put grooves in the bore, and the barrel still has to be chamber reamed, bolt lug recesses cut, etc., the bore is by no means finished.The third, and in my opinion the overall best, method is called hammer forging, and it is very different. It is the most modern method. In hammer forging, the entire barrel is shaped in one quick process, bore, rifling, chamber, in some cases even the finished exterior form. The barrel starts as a piece of steel called a billet. The billet is a short, thick piece of steel with a large hole through the length of it. It is placed onto a tool called a mandrel, and the mandrel is the key to the whole operation. It has all the interior features of the barrel machined onto it, bore, rifling, chamber, everything. In other words it is similar to a finished barrel turned inside out. The mandrel with the billet on it is loaded into the hammer forge, and hydraulic hammers close up on the billet, striking thousands of times per minute, and work the billet, with the mandrel inside it, into the shape of a rifle barrel. When the operation is complete, the mandrel is removed, and the bore usually needs nothing else done to it. The barrel has whatever needs to be done to the exterior completed, and it is finished. Stay safe, friends.