Tests done!

[davv]

Yay, all my finals are done... for now. So now I can focus on doing very little until mid-January.

(Though I always will be doing something, even if that is writing weird things here, or making fractal renderers for fun.)

Somehow, I feel quite tired, though. Perhaps I should try not to burden my mind too much, since I have been burdening it quite a bit on the preparations for the finals. But that's easier said than done, because I like using my mind. We'll see how that goes - I know I should rest a bit, but the tiredness itself makes it harder to do the right kind of decisions, so there's some element of positive feedback there.

As for the subjects I had tests in, there were three: mathematics (mostly calculus), numerical analysis (including plain old differential equations), and basic scientific programming. I think I did best at programming, then mathematics, then analysis. Amusingly enough, it was not so much the computer parts of numerical analysis that got me down as the mathematical ones (e.g. analytically solving linear and separable ODEs). I know the logic, but knowing the logic isn't enough: I also have to practice it, and I hadn't done that enough.

The only error I think I made on the programming thing was thinking that a Python dict is a primitive and so clones when you do an assignment of the type x = y. Call-by-sharing languages that exempt some types (but not all) from being references go against my aesthetics (for inconsistency purposes - but I've said so before) and I suppose it's only fitting that what snagged me was that particular feature :)

Comments

[lhexa]

Personally, I prefer to rest for about a week before final exams, not touching the exam material except for skimming it a little before the test. I've found that the relaxation helps more than the last-minute learning.

Hmm, over here "analysis" is a term for a more abstract sort of calculus, and your version would be called "mathematical methods" or "numerical methods", depending on how programming-intensive it is.

Were the linear ODEs you had to solve single- or multiple-variable? Linear equations do become easy after enough practice, but the multi-variable ones can get very confusing if you don't know much linear algebra. As for separable ODEs, from what I remember the main difficulty is recognizing whether or not a given equation is separable at all...

It probably won't surprise you to hear that linear and separable ODEs are vanishingly rare in advanced (and non-approximative) physics. Linear equations are mainly useful as an approximative tool, as ODEs are roughly linear when expanded around an equilibrium point. As it happens, linear PDEs are fairly common (most of E&M is linear, for example), but they're much, much harder than linear ODEs.

[davv]

Yeah, numerical methods is probably the more accurate term.

I have a similar relaxation strategy for the last day prior to each exam, and I do think the relaxation helps. But I feel a bit like there's not enough time to take a whole week to rest. (Perhaps that's what the others feel when they try to learn things on the very last day, too.)

They were single-variable. I think the problem was that they didn't really cover linear ODEs in lecture, and since I had other things to read on, I didn't check whether they would be on the tests. I discovered that they were about a month or so before the actual final, and that simply wasn't enough to properly practice.

I'm not surprised that they are rare. Actually, one of the professors made a remark of this nature. He said that in the kind of physics problems we solve by pen and paper, we'd usually encounter perfectly spherical cows, perfectly parabolic trajectories, etc... but that numerical methods made it possible to generally solve much more realistic problems without have to assume away too much.

Unless I take it manually (i.e. outside of the plan for the subcategory I've chosen), I won't encounter PDEs until a year or two from now. I'm curious about how to solve them, though, because I've heard doing so is pretty useful in control and regulation problems.

[lhexa]

In physics, at least, the method for solving linear PDEs is to find a convenient basis, and then expand the general solution in terms of this basis. Two of the best attributes for elements of this basis are:

First, being separable, so that, for example, a basis element obeys A(x,y,z) = B(x)C(y)D(z). For linear PDEs, at least, this allows you to to break apart the PDE into several ODEs, but at the cost of adding parameters (so you don't have one ODE to solve, you have a whole class of them). For instance, in spherically symmetric quantum mechanical systems, the parameters are energy, total angular momentum, and the z-component of angular momentum. Requiring convergence or periodicity of the functions will impose conditions on the parameters -- this is the origin of quantization, as it happens.

Second, having a trivial time-evolution equation. Basis elements in quantum mechanics tend to be eigenvectors of the Hamiltonian operator (that is, they are states with a definite energy), so their evolution is just harmonic oscillation. The full solution (with given initial conditions) will be neither separable nor harmonic, since these qualities break down when you start adding basis functions. But you can get around this by: expanding your initial state in terms of basis functions (this is usually some integral expression you have to derive); evolving the basis functions, in their trivial manner; recovering the evolved initial function by adding together the evolved basis functions. In lots of cases certain basis functions will have outsized importance (e.g. the ground state in QM), so you can cut corners.

This explanation is weighed toward what we do in physics, mind you.