Introduction to Electrodynamics Week 1

[unfocusedg...@gmail.com]

Welcome to the first week of Electrodynamics. Unless there is significant objection we will spend this week quickly reviewing vector calculus by going through all of Chapter 1. Hopefully there shouldn't be too many difficulties. 

I'll try to post a new weekly thread every Sunday evening/Monday from now on.

[sean.ma...@gmail.com]

Have you posted the thread?

Is there any specific words you'll use in the OP so we can just search the catalog for the thread? 

[unfocusedg...@gmail.com]

Hi, this is the thread. I don't think 4chan boards are a great environment to do stuff like this (high likelihood of disruptive posts ruining the threads, high chance of the thread 404ing in a few days, generally poor atmosphere for actual study). It'd just be lost in the sea of immature garbage.

[unfocusedg...@gmail.com]

I wish the formulae for the gradient/divergence/curl/Laplacian in cylindrical/spherical coordinates were easier to remember. Annoying to have to look them up...

[Thomas Lane]

The gradient and the laplacian are easy to remember, in my experience. But I use them about every week in other classes. Once you do a few problems, it just becomes second nature.

The divergence looks similar to the laplacian (at least that's how I will remember it). Although, every time I need it, I will probably derive the curl in spherical from the matrix determinant trick.

Spherical coordinates are too weird to have a good intuition about what the gradient, etc. will look like a priori.

[unfocusedg...@gmail.com]

I guess if one really needs to use the curvillinear forms much one will end up learning them naturally anyway, so not much point in memorising them.

If anyone wants to see vector calculus done rigorously (via Cartan's theory of differential forms) without having to learn differential geometry, I highly recommend the concise treatment in Rudin's Principles of Mathematical Analysis, which devotes a chapter to this subject. The (2<->3 dimensional) divergence theorem and the (1<->2 dimensional) Stokes theorem turn out to just be particular cases of the general Stoke's theorem which relates integrals between d and d+1 smooth (or more generally, C^2) manifolds. I think it generalises to more general manifolds but I've never had a need to learn integration on manifolds. 

Perhaps surprisingly, the development of the machinery of differential forms and the proof of the generalised Stokes' theorem (for C^2 manifolds at least) is actually quite straightforward once one has decided on the correct definitions. The most difficult point is the proof of the change of variables formula for integration (ie. integration by substitution), which is actually quite non-trivial and wasn't rigorously proved until the early 1800s.

Incidentally, the result that a vector field is curl free/irrotational (see section 1.6.2 of Griffiths) if and only if it is a gradient of a potential (eg. gravitational, electrostatic), and the result that a vector field is divergence free/solenoidal if and only if it is a curl of a certain vector field (eg. the stream function for an incompressible fluid) are both special cases of a more general theorem which are also proved in the above text. Though not the most general condition, this theorem holds in particular for R^3, spheres, cubes, plus importantly, all domains smoothly (or C^2) diffeomorphic to these. This covers most conceivable possibilities. 

A result from this chapter that I did not previously know of is the Helmholtz theorem which loosely states that a vector field is uniquely determined by its divergence and curl assuming that it tends to zero as we move away to infinity (which is a reasonable assumption for electric/magnetic fields).  Note that this assumption is indeed necessary, as there exist non-trivial fields with both zero divergence and curl (but these don't tend to zero towards infinity). In fact, we can find a formula recovering the field from its divergence and curl, which is given and proved in Appendix B. . For this result we need to assume that the divergence and curl divided by r^2 tend to zero as r tends to infinity, ie, are of order o(r^2). Typically I think we'd expect the electric/magnetic vector field strength to be roughly proportional to 1/r^2 at large distances from the charged/magnetised bodies I think, so the curl and divergence should be of order O(r^3), so it should be reasonable to assume this condition. It seems that this can be generalised to the more general Hodge decomposition for differential forms but I do not know the details.

[unfocusedg...@gmail.com]

I really can't stomach the proof of the Helmholtz theorem in the textbook though, due to the fact that the order of application of the integral and Laplacian at the bottom of page 555 and the top of page 556 are swapped without justification. I think this is really one of those times where care is needed, due to the presence of poles and the fact that we end up with a distribution (Dirac delta) in the integrand instead of an actual function. It seems to me that the fact that this yields the right answer is just good fortune more than anything at this point.

In general I think the treatment of the relationship between the Dirac point distribution with the vector field r/|r|^3 or the potential 1/r in this level of literature is sloppy... I wonder if anyone else can explain this well or direct me to somewhere that does? I'll try asking on StackExchange for now I guess.

Edit: hmm, looking at the proof www.ph.ed.ac.uk/~rhorsley/SI10-11_t+f/lec19.pdf, on page 2, it essentially boils down to showing that $v(r) = -\frac{1}{4\pi } \int \frac{a_i(r')}{|r-r'|} dr'$ solves the Poisson equation $ \nabla ^2 v_i = a_i$, ie. showing that $-\frac{1}{4\pi |r-r'|} $ is the Green's function for the Poisson equation. I guess that should be easy to find a real proof of.

[Thomas Lane]

Not sure if this is the level of sophistication you're asking for but whatever.

The Dirac Delta function in one dimension is an infinitesimally wide, infinitely tall (sometimes "bigger") spike that is zero everywhere else. The integral from negative infinity to infinity of this function f(x) is exactly 1. When you multiply any function by this you're going to get a "bigger" (maybe "smaller") spike that will contain an area equal to f(0). When you shift the delta function along the axis, it will still have area of 1 under the spike, and if you're multiplying by f(x), you're integral will become exactly f(a) (where a is the new location of the infinitesimal spike).

If you think about trying to do this in three dimensions, you can easily see what he is talking about in the book. I do agree, it is a little sloppy (at least compared to the explanations I've gotten about the DDF in the past).

[unfocusedg...@gmail.com]

Yeah, the definition of the delta function is fine, but when these applied texts use them in manipulations like this I think it's important to realise that they're basically just a sketch proof. 

For example, as I mentioned before, Helmholtz's theorem can be proved using the relationship to solution of the Poisson PDE equation via Green's functions. Showing that the Green's function representation actually gives a solution can be sketched using delta functions in a few lines, but to prove this rigorously takes a bit more work: 

https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&cad=rja&ved=0CE0QFjAF&url=http%3A%2F%2Fmath.mit.edu%2F~jspeck%2F18.152_Fall2011%2FLecture%2520notes%2F18152%2520lecture%2520notes%2520-%25207.pdf&ei=ypAbUZ2uOaGx0AXrvYCQCw&usg=AFQjCNHsOipaw_TPz0QCV_eLgjWdnZbN-w&sig2=JBwEo9VlGoicFGiZU66nDQ&bvm=bv.42261806,d.d2k  (Theorem 1.1)

[Thomas Lane]

The author was trying to provide motivation to talk about the Dirac delta function. The divergence of \rhat / r^2 does behave like the DDF (I'm only an undergrad in math who is currently taking analysis 1 so I can't go any farther than this). I agree the math is extremely important, (and I hate to say this) but this is an undergraduate physics text; it's hard to expect rigorous math here.

[unfocusedg...@gmail.com]

Yeah, I know. I just think there should be a word of warning that it's only a sketch :)

Actually the proof I linked isn't very hard and could have just have well been included in the book instead. The above proof essentially shows that swapping the order is valid provided the second order derivatives exist and are continuous.