Reunion - Lesson 7 - New Is Old - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Act III

N. New Is Old

N.5. Reunion

Let’s take a minute to remind ourselves of what we did in this chapter.

1.We began with a dialogue in which Author disclosed that he did not feel he had done this subject justice. Nevertheless, this chapter will serve as a bridge to get us where we need to go.

2.This led us into a tour of the conceptual origins of multivariable calculus, where—

Mathematics: WHAT’S MULTIVARIABLE CALCULUS?

Reader: Hey, I didn’t know you were allowed in here.

Author: Yeah, Math. Do you want us to finish the chapter or not?

Mathematics: I WANT YOU TO ANSWER ME. WHAT’S MULTIVARIABLE CALCULUS? WHAT’S THIS NEW PART OF ME YOU INVENTED WHILE I WAS AWAY?

Author: Why don’t you go read the chapter while I’m finishing it up?

Mathematics: . . . OKAY.

(Mathematics flips back and begins reading the chapter.)

Author: As I was saying, we took a tour of the conceptual origins of multivariable calculus, and we saw that many of its definitions spring from the desire to build new ideas out of nothing but old ideas.

3.We attempted to differentiate a machine of the form m(x) ≡ (f(x), g(x)) by doing the same things we would have done in single-variable calculus. Eventually we got stuck, but we found that we could get unstuck if we defined the addition of lists to be slot by slot: (a, b) + (A, B) = (a + A, b+ B). Being unstuck, we eventually got stuck again. To get unstuck the second time, we defined the multiplication of a number by a list to be multiplication slot by slot:

c · (x, y) = (cx, cy)

Having made these two choices, we found that

This meant our new “one in, two out” machines could be differentiated by using only familiar single-variable calculus in each slot.

4.We found that a similar story held for “one in, n out” machines of the form

m(x) ≡ (f1(x), f2(x), . . . , fn(x))

5.Because of this, we found that we could bring all of our hammers for computing derivatives into this new multivariable world.

6.We found that a similar story was true for integration. Namely,

As before, simply perform the familiar operations of single-variable calculus in each slot.

7.Because of this, we found that we could bring all of our anti-hammers for computing integrals into this new multivariable world.

8.We then turned our sights to “two in, one out” machines. We found that we could now define two different derivatives: one for each input slot. Textbooks call these “partial derivatives.” These partial derivatives were simply single-variable calculus all over again. For example, the partial derivative with respect to x is whatever we would get from pretending all the variables except x were constants, and computing the derivative in the familiar way.

9.We found that a similar story held for “n in, one out” machines.

10.Using our infinite magnifying glass and some simple visual reasoning, we derived the formula

and its generalization to n dimensions

11.Repeatedly in this chapter, whenever we found ourselves faced with a new idea, we found that it was not new after all, but had instead been cobbled together from old ideas, plus the occasional insignificant change to make everything make sense.

Author: Okay, Math. Are you happy now?

(Mathematics is reading the chapter and doesn’t hear Author.)

Author: Mathematics!

Mathematics: AAAH! YOU STARTLED ME.

Author: I’m “done” writing the chapter. Though I’m not exactly hapy with it. It’s horrible. See that typo I just made? Either way, we’re done now. Feel like you understand the chapter?

Mathematics: WELL, I ONLY HAD A CHANCE TO SKIM IT. LET ME SEE IF I’VE GOT THE IDEA STRAIGHT: WE’RE DOING CALCULUS JUST LIKE WE ALWAYS HAVE, BUT NOW OUR INTEGRALS AND DERIVATIVES ARE ACTING ON WEIRD NEW TYPES OF MACHINES, LIKE ONES THAT EAT A NUMBER AND SPIT OUT A VECTOR, OR EAT A VECTOR AND SPIT OUT A NUMBER?

Reader: Yep. That was basically all there was to it. Nothing much new.

Mathematics: AND IF I UNDERSTAND CORRECTLY, THESE “VECTOR” THINGS ARE JUST MACHINES THEMSELVES, RIGHT?

Author: What? No! Okay, hold on. We’re already hanging off the end of the Reunion, and this seems like it might take a while. Jump into the next interlude with me, you two.

(Author slams the door as he leaves the chapter.)