Kevin Knudson: Welcome to My Favorite Theorem. I’m Kevin Knudson, professor of mathematics at the University of Florida, and I’m joined by my cohost.
Evelyn Lamb: I’m Evelyn Lamb. I’m a freelance writer currently based in Paris.
KK: Yeah, Paris. Paris is better than Gainesville. I mean, Gainesville’s nice and everything.
EL: Depends on how much you like alligators.
KK: I don’t like alligators that much.
KK: This episode, we’re thrilled to welcome Amie Wilkinson of the University of Chicago. Amie’s a fantastic mathematician. Say hi, Amie, and tell everyone about yourself.
AW: Hi, everyone. So Kevin and I go way back. I’m a professor at the University of Chicago. Kevin and I first met when we were pretty fresh out of graduate school. We were postdocs at Northwestern, and now we’ve kind of gone our separate ways but have stayed in touch over the years.
KK: And, let’s see, my son and your daughter were born the same very hot summer in Chicago.
AW: Yeah, that’s right.
KK: That’s a long time ago.
AW: Right. And they’re both pretty hot kids.
KK: They are, yes. So, Amie, you haven’t shared what your favorite theorem is with Evelyn and me, so this will be a complete surprise for us, and we’ll try to keep up. So what’s your favorite theorem?
AW: Fundamental theorem of calculus.
EL: It’s a good theorem.
KK: I like that theorem. I just taught calc one, so this is fresh in my mind. I can work with this.
AW: Excellent. Probably fresher than it is in my mind.
EL: Can you tell us, remind our listeners, or tell our listeners what the fundamental theorem of calculus is?
AW: The fundamental theorem of calculus is a magic theorem as far as I’m concerned, that relates two different concepts: differentiation and integration.
So integration roughly is the computation of area, like the area of a square, area of the inside of a triangle, and so on. But you can make much more general computations of area like Archimedes did a long time ago, the area inside of a curve, like the area inside of a circle. There’s long been built up, going back to the Greeks, this notion of area, and even ways to compute it. That’s called integration.
Differentiation, on the other hand, it has to do with motion. In its earliest forms, to differentiate a function means to compute its slope, or speed, velocity. It’s a computation of velocity. It’s a way of measuring instantaneous motion. Both of these notions go way back, to the Greeks in the case of area, back to the 15th century and the people at Oxford for the computation of speed, and it wasn’t until the 17th century that the two were connected. First by someone named James Gregory, and not long after, sort of concurrently, by Isaac Barrow, who was the advisor of Isaac Newton. Newton was the one who really formalized the connection between the two.
EL: Right, but this wasn’t just a lightning bolt that suddenly came from Newton, but it had been building up for a while.
AW: Building up, actually in some sense I think it was a lightning bolt, in the sense that all of the progress happened within maybe a 30-year period, so in the world of mathematics, that’s sort of, you could even say that’s a fad or a trend. Someone does something, and you’re like, oh my god, let’s see what we can do with this. It’s an amazing insight that the two are connected.
The most concrete illustration of this is actually one I read on Wikipedia, which says that suppose you’re in a car, and you’re not the driver because otherwise this would be a very scary application. You can’t see outside of the car, but you can see the odometer. Sorry, you can’t see the odometer either. Someone’s put tape over it. But you can see the speedometer. And that’s telling you your velocity at every second. Every instance there’s a number. And what the fundamental theorem of calculus says is that if you add up all of those numbers over a given interval of time, it’s going to tell you how far you’ve traveled.
AW: You could just take the speed that you see on the odometer the minute you start driving the car and then multiply by the amount of time that you travel, and that’ll give you kind of an approximate idea, but you instead could break the time into two pieces and take the velocity that you see at the time and the velocity that you see at the midpoint, and take the average of those two velocities, multiplied by the amount of time, and that’ll give you a better sense. And basically it says to compute the average velocity multiplied by the time, and you’re going to get how far you’ve gone. That’s basically what the fundamental theorem of calculus means.
KK: So here’s my own hot take on the fundamental theorem: I think it’s actually named incorrectly. I think the mean value theorem is the real fundamental theorem of calculus.
KK: If you think about the fundamental theorem, it’s actually a pretty quick corollary to the fundamental theorem.
KK: Which essentially just describes, well, the version of the fundamental theorem that calculus remember, namely that to compute a definite integral, “all you have to do” — and our listeners can’t see me doing the air quotes—but“all you have to do” is find the antiderivative of the function, we know how hard that problem is. That’s a pretty quick corollary of the mean value theorem, basically by the process you just described, right? You’ve got your function, and you’re trying to compute the definite integral, so what do you do? Well, you take a Riemann sum, chop it into pieces. Then the mean value theorem says over each subinterval, there’s some point in there where the derivative equals the average rate of change over that little subinterval. And so you replace with all that, and that’s how you see the fundamental theorem just drop out. This Riemann sum is essentially just saying, OK, you find the antiderivative and that’s the story. So I used to sort of joke, I always joke with my students, that one of these days I’m going to write an advanced calculus book sort of like “Where’s Waldo,” but it’s going to be “Where’s the Mean Value Theorem?”
AW: I like that.
KK: Whenever you teach advanced calculus for real, not just that first course, you start to see the mean value theorem everywhere.
AW: See, I think of the mean value theorem as being the flip side of the fundamental theorem of calculus. To me, what is the mean value theorem? The mean value theorem is a movie that I saw in high school calculus that was probably filmed in, like, 1960-something.
KK: Right. On a movie projector?
AW: Yeah, on a movie projector.
KK: A lot of our listeners won’t know what that is.
AW: It’s a very simple little story. A guy’s driving, again it’s a driving analogy.
KK: Sure, I use these all the time too.
AW: And he stops at a toll booth to get his ticket, and the ticket is stamped with the time that he crosses the tollbooth, and then he’s driving and driving, and he gets to the other tollbooth and hands the ticket to the toll-taker, and the toll-taker says, “You’ve been speedin.’ The reason I know this is the mean value theorem.” He says it just like that, “The mean value theorem.” I wish I could find that movie. I’m sure I could. It’s so brilliant. What that’s saying is if I know the distance I’ve traveled from A to B, I could calculate what the average speed is by just taking, OK, I know how much time it took. So that second toll-taker knows (a) how much time it took, and (b), the distance because he knows the other tollbooth, right? And so he computes the average speed, and what the mean value theorem says is somewhere during that trip, you had to be traveling the average speed.
AW: So, it’s sort of like I can do speed from distance, so if you took too little time to travel the distance, you had to be speeding at some point, which is so beautiful. That’s sort of the flip side. If you know the distance and the amount of time, then you know the average speed. Whereas the first illustration I gave is you’re in this car, and you can’t see outside or the odometer, but you know the average speed, and that tells you the distance.
KK: So maybe they’re the same theorem.
EL: They’re all the same.
AW: In some sense, right.
KK: I think this is why I still love teaching calculus. I’ve been doing it for, like, 25 years, but I never get tired of it. It’s endlessly fascinating.
AW: That’s wonderful. We need more calculus teachers like you.
KK: I don’t know about that, but I do still love it.
AW: Or at least with your attitude.
KK: Right. There we go. So this is actually, the fundamental theorem is just sort of a one-dimensional version. There are generalizations, yes?
AW: Yes, there are. That gets to my favorite generalization of the fundamental theorem of calculus, which is Stokes' Theorem.
AW: So what does Stokes’ Theorem do? Well, for one thing, it explains why π appears both in the formula for the circumference of a circle and in the formula for the area of the circle, inside of the circle.
KK: That’s cool.
AW: Right? One is πr2, and the other is 2πr, and roughly speaking, suppose you differentiate with respect to r. This is sort of bogus, but it’s correct.
KK: Let’s go with it.
AW: You differentiate πr2, you get 2πr. The point is that Stokes'’ Theorem, like the fundamental theorem of calculus, relates two quantities of a geometric object, in this case a circle. One is an integral inside the object, and the other is an integral on the boundary of the object. And what are you integrating? So Stokes' Theorem says if you have something called a form, and it’s defined on the boundary of an object, and you differentiate the form, then the integral of the derivative of the form on the inside is the integral of the original form on the boundary.
AW: And the best way to illustrate this is with a picture, I’m afraid. It’s a beautiful, the formula itself has this beautiful symmetry to it.
EL: Yeah. Well, our listeners will be able to see that online when we post this, so we’ll have a visual aid.
AW: OK. So Stokes' Theorem establishes the duality of differentiation on the one hand, which is like analysis-calculus, right, and taking the boundary of an object on the other hand.
KK: That’s geometry, right.
AW: And boundary we denote by something that looks like a d, but it’s sort of curly, and we call it del. And differentiation we denote by d. The point is that those two operations can be switched and you get the same thing. You switch those operations in two different places, you get the same thing. That duality leads to differential topology. I mean, it’s just… The next theorem that’s amazing is De Rahm’s theorem that comes out of that.
KK: Let’s not go that far.
KK: It’s remarkable. You think, in calculus 3, at the very end we teach students Stokes’ Theorem, but we sort of get there incrementally, right? We teach Green’s theorem in the plane, and then we give them the divergence theorem, right, which is still the same. They’re all the same theorem, and we never really tie it together really well, and we never go, oh, by the way, if we would unify this idea, we’d say, by the way, this is really just the fundamental theorem of calculus.
KK: If you take your manifold to be a closed interval in the plane. So this makes me wonder if we need to start modernizing the calculus curriculum. On the other hand, then that gets a little New Math-y, right?
AW: No, no, I think we should totally normalize the curriculum in this way.
KK: Do you?
AW: Yeah, sure. It depends on what level we’re talking about, obviously, but I’ve always found that, OK, so, I’m going to confess the one time I taught multivariable calculus to “regular” students — granted, this was ages ago — I was so irritated by the current curriculum I couldn’t hide it.
KK: Oh, I see.
AW: But I’ve taught, lots and lots of times, multivariable calculus to somewhat more advanced students, to honors students who might become math majors, might not. And I always adopt this viewpoint, that the fundamental theorem of calculus is relating your object — your geometric object is just an interval, and it’s boundary is just two points, and differentiation-integration connects the difference of values of functions at two points with the integral over the interval.
KK: Then that gets to the question of, is that the right message for everyone? I could imagine this does work well with students who might want to be math majors. But in an engineering school, for example. I haven’t taught multivariable in maybe 15 years, but I’m tending to aim at engineers. But engineers, they don’t work outside of three dimensions, for the most part. Would this really be the right way to go? I don’t know.
AW: First of all, it’s good for turning students who are interested in calculus, who are interested in math, into math majors. So for me, that’s an effective tool.
KK: I absolutely believe that.
AW: Yeah, I don’t know about engineering students. They really have a distinct set of needs.
AW: I mean, social scientists, for example, work regularly in very high dimensions, and I have taught this material to social scientists back at Northwestern, and that was also, I think, pretty successful.
KK: Interesting. Well, that’s a good theorem. We love the fundamental theorem around here.
EL: The best things in life are often better together. So one of the things we like to do on My Favorite Theorem is to ask our guests to pick a pairing for their theorem, a fine wine or tea, beer, ice cream, piece of music, so what would you like to pair with the Fundamental Theorem of Caclulus?
AW: Something like a mango, maybe.
EL: A mango!
AW: Something where you have this organic, beautiful shape that, if you wanted to understand it analytically, you would have to use calculus. So first of all, mango is literally my favorite.
KK: I love them too. Oh, man.
AW: Ripe mango. It has to be good. Bad mango is torture.
KK: This is one of the perks of living in Florida. We have good mangoes here.
AW: What I love about the mango is it’s a natural form that is truly not spherical. It’s a fruit that has this clearly organic and very smooth shape. But to describe it, I don’t even know.
KK: It’s not a solid of revolution.
AW: I don’t know why it grows like that.
KK: Well the pit is weird, right? The pit’s sort of flat.
KK: Why does it grow like that? That’s interesting. Because most things, like an avocado, for example, it’s sort of pear-shaped, and the pit is round.
AW: An avocado is another example of a beautiful organic shape that is not perfectly spherical. So yeah, and I love avocado as well, so maybe I could have a mango-avocado salad.
EL: Oh, yeah. Really getting quite gourmet.
KK: And this goes to the fundamental theorem, right? Because you have to chop that up into pieces, which, I mean.
KK: It’s sort of the Riemann sum of your two things.
AW: And they’re very hard, both of them are very hard to get the fruit out, reasonably difficult to get the fruit out of the shell.
KK: You know the deal, right? You cut it in half first and then you dice it and scoop it out, right?
AW: You mean with the mango, right?
KK: You do with an avocado, too. Yeah.
AW: You know, I’ve never thought to do that with an avocado.
KK: Yeah, you cut the avocado, take a big knife and just cut it and then split it open, pop the pit out, and then just dice it and scoop it out.
AW: Oh. I usually just scoop and dice, but you’re right. In the mango you do the same, but then you start turning it inside out, and it looks like a hand grenade. So beautiful.
KK: You do the same thing with the avocado, and just scoop it. See?
AW: That’s a really interesting illustration, too, because when you turn inside out the mango, you can see these cubes of fruit that are spreading apart. You sort of can see how by changing the shape of the boundary, you change radically the sort of volume enclosed by the boundary. Because those things spread apart because of the reversed curvature.
KK: Now I’m getting hungry.
EL: Yeah, that’s the problem with these pairings, right? We record an episode, and then we all have to go out to eat.
AW: Of course a more provincial kind of thing, a more everyday object, piece of fruit, would be, as you said, pear. That’s more connected to Isaac Newton.
EL: True, yeah.
KK: The apples falling on his head, yeah. Cool. Well, this was fun, Amie. Thanks for joining us. Anything else you want to add? Any projects you want to plug? We try to give everybody a chance to do that. What are you working on these days?
AW: My area is dynamical systems, which..
KK: Is hard!
AW: It’s hard, but it’s also connected very closely. It’s not that hard.
KK: Smale said it’s hard.
AW: It’s connected very closely to the fundamental theorem. I study how things change over time.
AW: So I’ve been helping out, or I don’t know if I’ve actually been helping, but I’ve been talking a lot with some physicists who build particle accelerators, and we’re trying to use tools from pure mathematics to design these accelerators more effectively.
EL: Oh wow.
AW: To keep the particles inside the accelerator, moving in a focused beam.
AW: It’s a direct application of certain areas of smooth dynamical systems.
KK: Very cool. You never know where your career is going to take you.
AW: It’s very fun.
KK: That’s part of the beauty of mathematics, you never know where it’s going to lead you.
EL: Thanks so much for joining us on My Favorite Theorem.
AW: Thank you for having me. It’s been a lot of fun.
KK: Thanks for listening to My Favorite Theorem, hosted by Kevin Knudson and Evelyn Lamb. The music you’re hearing is a piece called Fractalia, a percussion quartet performed by four high school students from Gainesville, Florida. They are Blake Crawford, Gus Knudson, Dell Mitchell, and Baochau Nguyen. You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at Kevin’s website, kpknudson.com, and Evelyn’s blog, Roots of Unity, on the Scientific American blog network. We love to hear from our listeners, so please drop us a line at firstname.lastname@example.org. Or you can find us on Facebook and Twitter. Kevin’s handle on Twitter is @niveknosdunk, and Evelyn’s is @evelynjlamb. The show itself also has a Twitter feed. The handle is @myfavethm. Join us next time to learn another fascinating piece of mathematics.