MAX EBB — I'
m not big on crossword puzzles. But at my age I need some sort of mental exercise to fend off age-related dementia, so after considering many options, I settled on my preferred brain-stimulating activity: Tutoring calculus at my local public high school. It was amazing how much I had forgotten, but after dusting off my old calculus textbook (and some remedial sessions with Salazar Khan and his online Academy) I was almost up to speed on the weekly assignments. Yes, this would keep me sharp if anything would. One afternoon I was happily explaining integration by parts to a kid who had the concept down but kept getting the wrong answers, mostly because he kept losing track of negative signs when integrating trig functions. "It's the reverse of the product rule," I said. "After that it's just an accounting problem, and you need to keep better track of each step. Forget that little square of blank space in the workbook. Here, take this pad of graph paper. Write big, show every step, keep as little info as possible in your head and put as much detail as you can on the paper." There was a break in our discussion as the student re-worked the problem as per my recommendations, with every step written out in a much larger size and with a much darker marking pen. That's when I heard a familiar voice coming from the opposite corner of the classroom. It took a few seconds to recognize, but it was Lee Helm, totally out of context here in the high school math classroom. "And like, what's really cool about integration," she explained with obvious enthusiasm for the subject, "is that you don't have to limit yourself to integrating over an interval. You can follow any arbitrary path through a scalar field, or even a vector field. You can even integrate a surface around a volume. And like, things get really interesting when you integrate in the complex plane, where there's an imaginary axis that has a factor of i, or, like, square root of negative one." "Lee!" I hailed from across the room. "What's with the complex plane? That's Euler's territory. This is calculus AB, and we're supposed to be in the world of Newton." "Sure, but this kid's good," She explained, gesturing to her female student, who appeared to be at least a year or two younger than the rest of the calculus class. "And what are you doing in the high Page 90 •
• June, 2019
Sir Issac Newton was an English mathematician, physicist, astronomer, theologian, and author (described in his own day as a "natural philosopher") who is widely recognized as one of the most inﬂuential scientists of all time, and a key ﬁgure in the scientiﬁc revolution, according to Wikipedia.
school?" I asked. "Don't you have an assistant teaching gig in grad school? You know, where there are students who might actually need help with surface integrals in complex space." "It's like, court-ordered public service," she sighed. "What about you?" I explained my aversion to crossword puzzles, but I had to ask again to get Lee to tell her story. "Second moving violation," she finally admitted. "The judge offered to keep it off my driving record if I did 20 hours of community service, and tutoring math seemed like the most painless way to get the ticket punched." "But Lee. You don't even have a car!" "I blew through another stop sign on my bike," she confessed. "And like, the intersection was totally clear. Totally. I checked for cross traffic, I checked for pedestrians. There was, like, nothing unsafe about it. "And cars roll through stop signs all
the time." Meanwhile, my student had finally got the right answer for the integration- byparts exercise and was ready for some guidance on the next problem. "Back to Newton," I said. "Where none of the numbers are imaginary." "You know, Newton was not right about everything," Lee reminded me. "He was dead wrong on ship resistance, and it took Euler to come along some 50 years later to straighten everything out." "Really?" said my student. "Newton was wrong?" "For sure," said Lee. "He had this ridiculous 'shock theory' of ship resistance. According to Newton, water could be represented as a bunch of tiny little balls of fluid that bounced off the bow of a ship, imparting their momentum and causing drag. He derived drag as density times area times v-squared times sinesquared of the incident angle to the flow. Off by a factor of almost two!"
The June 2019 issue of the West's premier sailing and marine magazine.