My hands are very tired. I turned in an EM project this Monday that I spent about 12 hours on the night and morning before, pretty much just typing (mostly exporting data from Maple to Gnuplot, to Acrobat, to Illustrator– this is a process I find myself going through way too often– I will pay cash to anyone who can show me how to automate this [ I thought of using DDE or some similar programmed solution, but I don’t have prodigious amounts of time to waste on learning Microsoft’s crappy technology] ). I better get an A for effort, at the least. I hate those projects; it seems they exist solely because most people score low on the tests in that class. Since I have no problem with my test scores, I just find them tedious and boring; even if I did need to boost my test scores, I don’t think I’d find those very useful, because they are too damn tedious for you to do well on! If you’ve ever worked a contiguous 4 hour period on one assignment, knowing that you’re not even half way done yet, and it’s due tomorrow, you know what I mean.

Back to the topic of the moment: the calculus of variations. And what a cool topic; if you’ve never heard of it before, I’m shocked. If you’ve heard of it before, and actually have an idea of what it is, and you’re not a hardcore mathematician, mechanical engineer, or physicist (or the appropriate type of student), I’d be shocked and pleased. I personally have found it to be one of those topics, like measures, that everyone (read I) have encountered, but never found a completely satisfactory explanation of. Either it is mentioned in a context which assumes you know what it is already (” and this is a standard result from the calculus of variation”, … ), or the explanation attempts to subsume too many technical details into itself (like, explaining measures as tools used in constructing generalized integrals– what the hell does that tell you, unless you know that measures are generalized volume metrics?)

So here’s an excellent introduction to the subject (caveat: this is only in my meaningless opinion, since I haven’t read more than 10 pages on the topic, much of which I’m not sure I understand) :

The calculus of variations is concerned with finding extrema and, in this sense, it can be considered a branch of optimization. The problems and techniques in this branch, however, differ markedly from those involving the extrema of functions of several variables owing to the nature of the domain on the quantity to be optimized. A functional is a mapping from a set of functions to the real numbers. The calculus of variations deals with finding extrema for functionals as opposed to functions. The candidates in the competition for an extremum are thus functions as opposed to vectors in R^n, and this gives the subject a distinct character. The functionals are generally defined by definite integrals; the set of functions are often defined by boundary conditions and smoothness requirements, which arise in the formulation of the problem/model.

Nice.. it introduces the calculus of variations, and also the concept of a functional. Now I have a ready answer for those people ( non-electrical engineers, of course — we use it for just about everything, so we better not bite the hand that convolves for us) who proclaim the δ function is not a function, so is mathematically insensible. It is a function, just a function that maps from the space of continuous functions to the real numbers in the following fashion: . With this definition, the δ(t) relation has to be a function. It would be interesting to see how the usual properties of the δ function follow from this.

Now I wonder, are there any interesting uses for functionals that operate solely on other functionals? What would a functional have to be like to be considered continuous? I can’t wait for function theory! On that note, I go back to minding my undergrad p’s and q’s.

Possibly relevant posts:

• UH’s second annual middle and high school math competition (2/20/2006)
• DNG + CW = OMG, Fun fun fun! (9/8/2007)
• Folland for the weekend (3/30/2007)