This is part of my series of commentary on the physics book The Theoretical Minimum.
I’m lumping this all together because they’re all math background.
I read through these sections without skipping anything even though I didn’t have to, because, you know, math; I can literally do this stuff in my sleep (by which I mean I have successfully done calculus after being up for more than 40 hours). But it was very interesting to read the authors’ crash course in single-variable calc — the angle they came from was informative, and I picked up a few pleasing pedagogical tricks that I may use with some of my own students at some point.
That’s basically all there is to differential calculus. (p. 36)
This made me howl in amusement. Not because the book is wrong — it’s not! And they did a great job on teaching calculus in ten pages! — but because I imagined telling my high school students that. (To be clear, for a book on physics that is aimed at people who know calculus but may need a refresher, it was excellently done IMO.)
The fundamental theorem of calculus is one of the simplest and most beautiful results in mathematics. (p. 50)
There are some tricks to doing integrals. One trick is to look them up in a table of integrals. Another is to learn to use Mathematica. But if you’re on your own and you don’t recognize the integral, the oldest trick in the book is integration by parts. (p. 55)
This perspective is fascinating to me, and I suspect it’s one of those mathematician/physicist divides that made me twitch in my physics lectures in undergrad (and my friends would all roll their eyes and laugh at me). Because I would never consider either tables or Mathematica to qualify as actually “doing” integrals!
I was also surprised at the weight given to integration by parts as “the” basic integration tool. To me substitution is the most basic (and is a backwards chain rule rather than a backwards product rule) and is the first go-to. Integration by parts feels to me on the level of partial fractions or trig substitutions — useful for its own specific subset of problems, but hardly the broad skeleton key the book seems to imply. I do wonder if this is another mathematics/physics difference — if the vast majority of integrals physicists deal with happen to fall into the type that are solvable by parts, it makes perfect sense that it would be considered the basic tool of the trade.
I liked the framing of an indefinite integral as a definite integral with a variable limit — I hadn’t seen it explained in quite that way before.
Also, I knew integration by parts came straight out of the product rule, but I’ve been doing it for so long that a layer of abstraction had built up and I’d forgotten! Always good to be reminded of these connections. :)
Exercise 4: Prove the product rule and the chain rule.
I know I’ve done these proofs before, but it’s been long enough that I didn’t remember how they went, so I figured why not. My blog is being difficult about Latex formatting for some reason, so I’ll just give the gist.
Product rule: Add and subtract the appropriate quantity, and it all falls out very nicely.
Chain rule: Realize that g + Δg = g (t + Δt). This is easy to intuit via a visualization of a t versus g graph, with points marked (t, g(t)) and (t + Δt, g(t + Δt)). The gap between g(t) and g(t + Δt) becomes Δg, and voila.