I have three exams today, and for Analysis, there's a possibility we'll have to recite either the Inverse Function Theorem or the Implicit Function Theorem.

Inverse Function Theorem

Suppose f is continuously differentiable over U, some subset of Rⁿ, for f: Rⁿ → Rⁿ. Also suppose that, for some a in E, f'(a) is invertible, and that b = f(a). Then:

1. There exist U, V open in Rⁿ such that a is in U, b is in V, f is one-to-one on U, and f(U) = V
2. If g = f^-1(V) = f^-1(f(U)), then g is continuously differentiable over V

Implicit Function Theorem

Suppose that for f:R^n + m → Rⁿ, and for E an open subset of R^n + m, f is continuously differentiable over E. Suppose also that for some (a, b) in E, f(a, b) = 0. Let A = f'(a, b) = (A_x A_y), and assume that A_x is invertible.

Then there exist U in R^n + m, and W in R^m, both of which are open, so that (a, b) is in R^n + m, and b is in R^m. In fact, for every y in W, there exists a unique x such that f(x, y) = 0.

If we let this x = g(y), then f(g(y), y) = 0, g:W → Rⁿ is continuously differentiable, and g'(b) = -A_x^-1 · A_y.

As has become habit while mulling over (mathematical) morsels upstairs, I turn my mind to other matters. This weekend — and I'm a little surprised I hadn't considered this sort of thing before — I thought it would be interesting to measure a couple of things:

The first two relate to coffee: how much time will I have passed in a coffee shop when I have gone as far as my education will take me (for the sake of optimism, let's suppose I end up completing a PhD), and, more generally, how many gallons of coffee (and tea — I love chai) will I have consumed by this point?

When I first posed this question to myself, I tried to make a few rough extrapolative estimates, using my current affinity for coffee and my average number of trips to Espresso Royale as guides, but I quickly lost my mental sense of scale, and I haven't yet put forth the energy into punching a few buttons on my calculator.

I can tell you, though, that the numbers will be quite large, indeed. Embarrassingly so, you might say.

The third and final quantity in which I am interested is the number of reams of plain white paper I'll have used for scratch. Unless I know precisely what I am doing — and believe me, at this point, such an occurrence is rare — I tend to use at least one side of a sheet per problem. I tend to toss one or two sheets per assignment before I am through, and I rarely use pencil, because it often turns out that a once-crossed out result is actually pretty close to the mark, whereas an erased result is lost forever.

Of course, there are other considerations here, too. If I make it to graduate school, there's a decent chance I'll have a blackboard or a whiteboard to my name, at least in-part, in which case much of my scratch will end up on the wall, with a large "DNE" not too far away.

I got a haircut

Truth. First I was all :D, but then I was all :/. The world did not end, but it came close to ending. I take with me the knowledge that a relative measure of my hair — specifically, "one-half of my current length" — is meaningless to the lady holding the clippers.

I thought I was going to write more, but I simply cannot.