What’s really memorable about this past week, for me, was a short collaboration between my alter egos the kid and the undergraduate. The undergrad was struggling with easy stuff – and the kid really, really wanted to get it right.
I had been struggling with circuit theory… specifically with RLC circuits. Yes, I solved them long ago in college – although, in fact, I didn’t actually study elementary differential equations until I was a graduate student. (I hadn’t had them before I transferred to Caltech as a sophomore… where they had been covered freshman year.) I had also never seen Laplace transforms until I was a graduate student. I learned both during the 1st course for which I was a TA.
And here I was, all bollixed up, unable to get what I expected.
On the other hand, I had gotten there after deciding that I needed to solve one of the simplest possible LRC circuits, rather than the more complicated ones to which I was trying to apply a slightly more sophisticated method.
Anyway, I woke up one morning dreaming about it, and determined to forget all the complicated stuff… just solve the second-order differential equation for current… and then solve the equation using Laplace transforms. In fact, I solved the voltage balance rather than the second-order ODE using Laplace transforms – and there is one little tricky detail….
A piece of cake. All I had to do was pay attention. Yes, I learned a couple of things. The first was about me and circuits: although I certainly know better, I have a tendency to think of the second-order equation for current as though it were a voltage balance. It isn’t remotely. Grrr.
The second was that if we do want to use a voltage balance and Laplace transforms, then the initial voltage on each capacitor looks for all the world as though the capacitor had been replaced by a constant voltage source.
Which I had read, and not understood. So going back to the beginning had cleared up one of the more advanced issues. Cool.
Okay, it was all elementary… but I’m still delighted to have gotten it all to work out. This is, after all, something my undergraduate is trying to make sense of. When you get confused, simplify the problem. Now I’m ready to tackle more complicated circuits.
Let me close by reviewing the performance of this blog. I should have done it a few weeks ago but I was playing a computer game instead.
From its inception in November 2007 to December 31, 2011… the blog had had 143,500 hits, almost 53,000 spam… and a total of 372 posts.
During 2011 alone, the blog got more than 50,000 hits – more than 4000 almost every month, more than 5000 three times… with a single-day maximum of 309 hits. (The top line is 6000, then we have 5000, and 4000.)
7 posts each had more than 1000 hits in 2011… 2 of them were new: compressed sensing and 5 card draw poker hands – and each is a runaway bestseller.
8 posts have over 2000 hits for all time… one would not expect that list to change quickly… 7 of those posts were the top 7 listed in December 2010… compressed sensing is the new kid on the block.
And with that, let me get about doing and/or writing mathematics.
Oh, I got a new book this week: “Topological Methods in Hydrodynamics”, by Arnold and Khesin….