# The Theoretical Minimum: Interlude 1, Spaces, Trigonometry, and Vectors + Lecture 2, Motion + Interlude 2, Integral Calculus

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.

Yay calculus!

### Notable Quotes

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)

Agreed!

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.

### Thinky Thoughts

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. :)

### Worked Problems

#### 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.

# The Theoretical Minimum: Lecture 1, The Nature of Classical Physics

This is part of my series of commentary on the physics book The Theoretical Minimum.

### Notable Quotes

The job of classical mechanics is to predict the future. (p. 1)

I love this.

The rule that dynamical laws must be deterministic and reversible is so central to classical physics that we sometimes forget to mention it when teaching the subject.  In fact, it doesn’t even have a name.  We could call it the first law, but unfortunately there are already two first laws — Newton’s and the first law of thermodynamics.  There is even a zeroth law of thermodynamics.  So we have to go back to a minus-first law to gain priority for what is undoubtedly the most fundamental of all physical laws . . . (p. 9)

As a number-lover, this sort of thing just makes me all kind of amused.

But there is another element that [Laplace] may have underestimated [when he said the laws of physics could theoretically predict the whole future]: the ability to know the initial conditions with almost perfect precision. […] The ability to distinguish the neighboring values of these numbers is called “resolving power” of any experiment, and for any real observer it is limited.  In principle we cannot know the initial conditions with infinite precision. […] Perfect predictability is not achievable, simply because we are limited in our resolving power. (p. 14)

This concept, I am keenly aware, is what makes my Russell’s Attic books science fiction.  My main character is only able to do the calculations she can on the world around her because I permit her to have indefinitely good resolving power.  It’s kind of a required secondary power for what she does.  And reading this section, it completely tickled me that it has a name!

### Thinky Thoughts

I’ve said before that I reduce all physics to doing math.  I felt like I was cheating a bit in this section, because saying a system is deterministic and reversible is the same as saying you can model it with a one-to-one function.  So I bopped along just thinking of the functional invertibility of the the rules, most of which I knew off the top of my head.

Sigh.  You can take the mathematician out of mathematics . . .

# The Theoretical Minimum: Preface

So. I’ve always wanted to learn physics.

I know this will sound bizarre to most people.  I went to freakin’ MIT — one of my physics professors as an undergrad won the Nobel Prize the semester after I was in his class.  But I’ve never felt I’ve had a good intuitive grasp of physics, nor studied it to a depth where I felt like I understood it.

A confession: Physics doesn’t come naturally to me.  It’s not intuitive.  (I always end up reducing it to mathematics, and then it makes sense.)  Because physics ran a bit against the grain of my brain, I didn’t study it heavily in college beyond my undergraduate requirements, and I always regretted that.  I’ve always wanted to know more, and felt a bit like the physics world was this fascinating enchanted universe but I’d only ever managed to have my face pressed up to the glass, catching glimpses without being able to be a true party to the wonder.

While I was down with cancer, a college friend of mine gave me a book called The Theoretical Minimum, by Leonard Susskind and George Hrabovsky.  He billed it to me as, “for people who have a decent foundational background and know calculus, but who were never able to study physics to the point they wanted to.”  I said, “Hey, that’s me!”

Well, I’ve been reading it, and it’s quite excellent!

To keep myself on track with it, I’m going to blog about a chapter every week or thereabouts.  I apologize if people find these boring — I’ll try to keep them pithy.  I started the book a while ago so hopefully I should be able to keep up with the posts  (in fact, I’m going to write a few of these and buffer them before posting, and then send them up every Saturday or so).

So, these posts will be some commentary on The Theoretical Minimum, whatever I feel like writing about it.  I shall start with the Preface, since it’s quite worth starting with:

### Notable Quotes

As it happens, the Stanford area has a lot of people who once wanted to study physics, but life got in the way.  They had had all kinds of careers but never forgot their one-time infatuation with the laws of the universe.  Now, after a career or two, they wanted to get back into it, at least at a casual level. (p. ix)

This is pretty much exactly me.  Also, I often regret that there aren’t more opportunities for hobby academics — thanks to online endeavors like Coursera and edX, that’s changing, happily, but it can sometimes be hard to find courses that are just that bit beyond the basics.  Especially for more theoretical scientific disciplines.

Okay, now next week — on to the physics!