Isomorphismes

W Gilbert Strang, a really excellent lecturer who sees teaching as central to his purpose in life, has made his complete calculus course available for free here. He has also summarised the basics of calculus, using simple words and examples, boiling the subject down to 2.5 hours here.

Calculus is not all there is to post-secondary mathematics. But it does symbolise “difficult! complicated! advanced!” for a lot of people who didn’t devote ≥hundreds of hours of their adult lives to mathematics. If you felt like you never “got” maths in school, watching and understanding 5 half-hour videos about a “complicated! scary!” subject with Sir Isaac Newton’s name attached to it could give you a huge boost in self-confidence.

I’ll try to say what I think calculus is about in even fewer words.

• Calculus is about zooming in on things.
• In some sense the subject is infinity. You zoom in to an infinitesimal level.
• It can feel downright spiritual.
• When you zoom in on a smooth curve to an infinite degree, you find that, “under the microscope”, the curve is actually straight.
• It’s like what would happen if you kept hitting “zoom” on a picture of Y=sin(X) on a graphing calculator.
  
   
• Straight things are really easy to analyse mathematically. Like a straight-line projection is much simpler than a noisy, seasonal, highly variable projection.
   
• The most surprising thing to come out of the rigorous study of infinitesimally zooming in is that area is the opposite of slope.
• In some sense this finding is like multiplication is the opposite of division.
  • multiplication ↔ area ↔ ”integral” ∫ ↔ “tabulate” or “tally up” , and
  • division ↔ slope ↔ ”derivative” ∂ ↔ “pulse”
• Saying that calculus is about derivatives ∂ and integrals ∫ is a little misleading. Saying that ∂ is about “slope” is not quite right either. Saying ∫ is about “area” is not quite right either! But just like the solar system model of the atom, I am lying to you at the start because the real story is too complicated.
  
• Have fun!

Imagine you were a wealthy writer — so wealthy that you could pay servants to look stuff up for you. Instead of drudging through tomes (or internet searches) to fact-check yourself, find original references, and so on. You just do the fun part: pontificate on paper.

Now let’s say after you have finished an essay, your servant / employee / virtual personal assistant comes back from his footnote research and tells you that statement #13 should be revised based on the best-known research on the topic. In fact, statement #13 is almost the reverse of the truth.

I can imagine things going one of two ways from here.

1. In the less interesting case, statement #13 is an offhand remark upon which little else in the essay depends. You correct yourself, modify some text directly before and after statement #13, and move on. The only neighbours of the concepts in statement #13 are the transition sentences directly before & after it on a 1-dimensional topological line.
   
    
2. In the more interesting case, what you said before statement #13 was meant to lead up to the exact statement you made. Perhaps #13 was a key point, or the thesis of the essay. And let’s further imagine that the text following statement #13 depended critically on the exact value of statement #13 being as you wrote it. When #13 is altered, the preceding text is no longer necessary and the succeeding text no longer works.

In an especially dire scenario, your PA’s research might overturn the worldview that led you to write the essay in the first place.

Like Holger Lippmann’s “Flower Circles 13”, changing one element renders the entire whole needful of alteration. Everything is so thoroughly enmeshed (see “complete graph” below for the neighbourhood relations) that no element of the text speaks in isolation.

That’s in distinction to the calculus, where smooth functions can be approximated by a differential.

In physicists’ language, due to tightly, globally connected topology, perturbations cannot be localised. Rather, the opposite: local perturbations cause global changes in the object.

OK, someone dared me. I’ll say it: Gestalt.

A beautiful depiction of a 1-form by Robert Ghrist. You never thought understanding a 1→1-dimensional ODE (or a 1-D vector field) would be so easy!

What his drawing makes obvious, is that images of Phase Space wear a totally different meaning than “up”, “down”, “left”, “right”. In this case up = more; down = less; left = before and right = after. So it’s unhelpful to think about derivative = slope.

BTW, the reason that ƒ must have an odd number of fixed points, follows from the “dissipative” assumption (“infinity repels”). If ƒ (−∞)→+∞, then the red line enters from the top-left. And if ƒ (+∞)→−∞, then the red line exits toward the bottom-right. So no matter how many wiggles, it must cross an odd number of times. (Rolle’s Thm / intermediate value theorem from undergrad calculus / analysis)

Found this via John D Cook.

(Source: math.upenn.edu)

I may have to take back what I wrote earlier. I need to find out more about the Radon-Nikodym concept of a derivative.

The LaPlace Transform is simpler than I thought. It’s just the continuous version of a power series.

Think of a power series


as mapping a sequence of constants to a function.

Well, it does, after all.

Then turn the ∑ into a ∫. And turn the x^k into a ln ( k ⨯ exp x ). Now you have the continuous version of the “spectrum” view that allows so many tortuous ODE’s to be solved in a flash. I wonder what the economic value of that formula is? It’s used in so many engineering applications.

Anyway, there is also wisdom to be had here. Thinking of functions as all being made up of the same components allows fair comparisons between them.

(If you really want to know what a power series is, read Roger Penrose’s book.

To summarize: a lot of functions can be approximated by summing weighted powers of the input variable, as an equally valid alternative to applying the function itself. For example, adding input¹  +  1/2 ⨯ input²  +  1/2/3 ⨯ input³  +  1/2/3/4 ⨯ input⁴ and so on, eventually approximates e^input.)

(Source: ocw.mit.edu)

The hallmark of a calculus course is epsilon-delta proofs. As one moves closer and closer to a point of interest (reducing δ, the distance from the point-of-interest), the phenomenon’s measure is bounded by something times ε, a linear error term. The bound comes from the continuity of the function, also defined in epsilon–delta terms.

So everything moves gradually, in a sense. There are no sudden jumps. But human affairs are characterized by jumping, leaping, gapping, sparking, snapping, exploding processes.

One studied example is stock prices. If terrible news hits about a public company, you won’t be able to sell your shares for the previous price minus epsilon. You’ll have to unload a gap or a yawn lower. Not that it was ever possible to trade in arbitrarily small δ intervals anyway. The smallest increment on the NYSE is $.01 (it used to be ⅛ of a dollar), which by infinitesimal standards is huge.

The Continuum

Speaking of infinitesimal standards, I need to digress for a few paragraphs so my point will make sense to all readers. Real numbers ℝ — any number you can construct with infinity decimal places, so essentially any number that most people consider a number at all — are thick, dense, an uncountable thicket. They are complete.

“The Reals” ℝ are made up of rational ℚ and irrational ℚᶜ numbers.

Rational numbers ℚ are ratios of regular counting numbers, 1, 2, 3, ℕ, etc., and their negatives −ℕ. However the rational part ℚ of the reals ℝ — the part that’s easy to conceive and talk about and imagine — is a negligible part of the real number line.

The irrational part ℚᶜ is further divided into algebraic 𝓐 and transcendental 𝓐ᶜ parts. Again the algebraic part 𝓐 is easier to explain and is, literally, negligible in size compared to the transcendental part.

Algebraic numbers 𝓐𝓐 are the x’s that solve various algebraic equations, like x²=2.

Whatever number x you square to get 2, is an algebraic number 𝓐. We invent a symbol √ and put it in front of the 2 symbol to express the number we’re talking about. Although there is no such symbol to express the number x that solves x² + x = 2 — square this number, then add itself to the result, and you get two — that is also an algebraic number.

Now add in all other finite-length equations with integer or fraction coefficients. That’s a lot of equations. Their solutions constitute the algebraic numbers 𝓐. But like I said above, 99% of the real numbers — those simple things you learned about in 3rd grade when they taught you the decimal system — are NOT IN THERE.

(99% of infinity, what am I talking about?  It doesn’t make sense, I know, just work with me here.)

Transcendental

OK so now I have gotten to these hard-to-describe numbers called transcendental 𝓐ᶜ. The black part in the picture above. It took me a few paragraphs just to sloppily say what they are. If you have never thought about this issue before it might take you hours to wrap your head around them.

But it’s these transcendental numbers 𝓐ᶜ— can’t be assembled without an infinitely long equation — which essentially make calculus work. Calculus depends upon the real numbers ℝ and continuity therein, and without this thick, dense, impenetrable subset 𝓐ᶜ called transcendentals, its theorems would be unprovable and illegitimate.

I don’t know about you, but I haven’t seen any transcendental numbers around lately! Other than e and π, I mean. Despite transcendental numbers being the most numerous, only a few are known, most based around e and π. That’s right, these are the largest exclusive subset of the real numbers, and we don’t really know that many of them. We use them in proofs but not by name. Just knowing that they’re there ensures that calculus works.

But in the real world, you can’t buy e/3 eggs.  That, among other reasons, means you can’t optimize — even in principle — a purchasing decision at the grocery using calculus. (Maybe you don’t think you would be using calculus anyway, but the economic theorists treat you like a gas particle dispersing in the room — and while the particle doesn’t think it’s using calculus to decide where to move, it obeys those laws. So they are relevant somewhere, contrary to the title of this post.)

Spiky

So calculus works in gas diffusion and solving various states of atoms / molecules via Schrödinger equations. But what about us people?

Here is a topographical picture of where people live. Notice that there is a lot of spikiness. Sudden jumps.

For a long time there’s no people because you’re in the middle of Nevada, and then — all of a sudden — Vegas! Holy cow there are people EVERYWHERE. Flowing in and out at a phenomenal rate. But there is zero flow and zero inhabitants just a few miles away. Molecules don’t behave like that.

Here is another picture — got it out of the same book, which my girlfriend is reading — of economic output by region.

Again, much spikiness.  Not much calculus. Discontinuous outputs. Maybe that is how we are. Maybe calculus doesn’t work on us.

The chief triumph of differential calculus is this:

Any nonlinear function can be approximated by a linear function.

(OK…pretty much any nonlinear function.) That approximation is the differential, aka the tangent line, aka the best affine approximation.  It is valid in only a small area but that’s good enough. Because small areas can be put together to make big areas. And short lines can make nonlinear* curves.

In other words, zoom in on a function enough and it looks like a simple line. Even when the zoomed-out picture is shaky, wiggly, jumpy, scrawly, volatile, or intermittently-volatile-and-not-volatile:

Moreover, calculus says how far off those linear approximations are. So you know how tiny the straight, flat puzzle pieces should be to look like a curve when put together. That kind of advice is good enough to engineer with.

It’s surprising that you can break things down like that, because nonlinear functions can get really, really intricate. The world is, like, complicated.

So it’s reassuring to know that ideas that are built up from counting & grouping rocks on the ground, and drawing lines & circles in the sand, are in principle capable of describing ocean currents, architecture, finance, computers, mechanics, earthquakes, electronics, physics.

(OK, there are other reasons to be less optimistic.)


 

* What’s so terrible about nonlinear functions anyway? They’re not terrible, they’re terribly interesting. It’s just nearly impossible to generally, completely and totally solve nonlinear problems.

But lines are doable. You can project lines outward. You can solve systems of linear equations with the tap of a computer.  So if it’s possible to decompose nonlinear things into linear pieces, you’re money.

Two more findings from calculus.

1. One can get closer to the nonlinear truth even faster by using polynomials. Put another way, the simple operations of + and ×, taught in elementary school, are good enough to do pretty much anything, so long as you do + and × enough times. 
   
   
2. One can also get arbitrarily truthy using trig functions. You may not remember sin & cos but they are dead simple. More later on the sexy things you can do with them (Fourier decomposition).

This is for my homies taking Calculus.

U-substitution is the opposite of the Chain Rule.  Integration by parts is the opposite of the Product Rule.

Don’t believe me?

Take u and v to be the left and right parts of some formula.

Now switch some symbols around and you’ve arrived at the formula for Integration by Parts.

By the way, I normally use L and R for the left and right groups of symbols when I’m teaching Product Rule.  Here I just used u and v because that’s probably how you’ve seen the formula be written.