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PROFESSOR: So now, let me make the first of our technical asides, which you can skip over if you're not interested in the mathematical details or for those of you that have a higher-level, say, upper-level-undergraduate or even graduate-level understanding of transport phenomena and fluid mechanics. I'd like to show you some of the equations that are behind the results that I've been quoting in all the lectures. So in particular, let's derive the Wells curve. So part of that was a theory of drop settling. Here, I will quote a certain result, because the derivation would be a lot longer. Actually, for that, you could refer to my online class 10.50x, which is that if you have a droplet or a particle of a radius R and it is settling under gravity-- so it has a mass m, and the gravitational force is m g, where g is the gravitational acceleration-- then there is a flow of fluid around this object. And relative to the moving object, the flow is going the other w...

PROFESSOR: So now that we've talked about the different dynamics of droplets in the air, we can think about how different types of pathogen can leverage those droplets to transmit from one person to another through respiration through the air. So let's begin with bacteria. So there are many different kinds of bacteria. The typical size of a single bacterium is on the order of several microns. So let's just say, 1 to 10 microns. On the other hand, bacteria can also exist in colonies or larger structures. And that does determine, to some extent, what kinds of droplets can transmit those bacteria. So let's begin with an example of a large-drop bacteria, which is typically going to be found in large drops. And that would be the bacteria that causes strep throat, the streptococcus. So the streptococcus bacteria, which is shown here, has a typical size that's around two microns. So it's on the smaller end. And it forms chains and even larger colonies the...

Three-dimensional case. Now, in the future homework, you will be doing the equivalent of this calculation here with the Laplacians-- it's not complicated-- so that you will derive with the current is. And the current must be a very similar formula as this one. And indeed, I'll just write it here. The current is h bar over m, the imaginary part of psi star. And instead of ddx, you expect the gradient of psi. That is the current for the probability in three dimensions. And the analog of this equation, d rho dt plus dj dx equals 0, is d rho dt plus divergence of j is equal to 0. That is current conservation. Perhaps you do remember that from your study of electromagnetism. That's how Maxwell discovered the displacement current when he tried to figure out how everything was compatible with current conservation. Anyway, that argument I'll do in a second so that it will become clearer. So one last thing here-- it's something also-- you can check the units he...

GILBERT STRANG: OK? Now, finally, a nonlinear equation. Growth, but it's-- the growth is cut off by competition-- slowed down my competition. Let me show you the equation. In a way, it's the simplest nonlinear equation I could think of. We have the usual dy/dt equal ay. That would give exponential growth. The growth would go on forever. But then, as y gets large-- if we think of y as the population, so the growth is exponential when the birth rates higher than the death rate, and we grow. We have-- and that's what's happening. But our world population is not growing pure exponentially, forever. And this is a simple term, a simple model not, very accurate, but it's the right place to start. For competition, when you have too many people, somehow the multiplying y times y gives you a number of interactions of all the y people with themselves. And those interactions-- that competition slows down the growth, and so the coefficient there is negative. We wan...

PROFESSOR: Now, what do we call these things-- pan out? What do you call these things-- pan out and to make up? To pan out, to make up. We call these things-- AUDIENCE: Phrasal verbs. PROFESSOR: Phrasal verbs. Phrasal verbs, OK? A verb usually has-- phrasal verbs usually have one verb and one particle, or it could have two more-- one particle and one preposition at the end. So what's the difference between a particle and a preposition? Does anybody-- what's a preposition? Give me some examples of prepositions. Prepositions. AUDIENCE: In. PROFESSOR: In. AUDIENCE: Of. PROFESSOR: Of. Of-- O-F, of. AUDIENCE: From. PROFESSOR: At, from-- OK, those are prepositions, right? Out, up-- these kind of look like prepositions, but they actually have some meaning. These are called adverbial particles. Adverbs-- they have some meaning. So now, if you turn to this handout, which is page 65 in your course packet. Page 65. Do you see where we left off last time? OK, this sentence-- ...

Now why does this matter? And I'm going to tell you an example of why this matters. I'm talking about delocalization to stabilize. And I'm talking about how the actual structure of a molecule isn't this rigid two bonds here, one bond there, that's not what that says, but it was in ozone. But it's this shared equivalent bonds where the electron is shared and the whole system is lowered. Now, there is one molecule where this is very much important and dictates all of its chemical behavior. And that's benzene. And back in the day, in the 1930s, oh, by the way, Pauling, electronegativity scale. we talked about him on Wednesday. But back in the day, they didn't know what benzene was. They could kind of measure some things. But none of it worked. And Pauling wrote a number of structural formulae have been proposed for benzene, but none of them is free from very serious objections. The way they wrote stuff. Look at these shapes for benzene. If we ...

GILBERT STRANG: Moving now to the second half of linear algebra. It's about eigenvalues and eigenvectors. The first half, I just had a matrix. I solved equations. The second half, you'll see the point of eigenvalues and eigenvectors as a new way to look deeper into the matrix to see what's important there. OK, so what are they? This is a big equation, S time x. So S is our matrix. And I've called it S because I'm taking it to be a symmetric matrix. What's on one side of the diagonal is also on the other side of the diagonal. So those have the beautiful properties. Those are the kings of linear algebra. Now, about eigenvectors x and eigenvalues lambda. So what does that equation, Sx equal lambda x, tell me? That says that I have a special vector x. When I multiply it by S, my matrix, I stay in the same direction as the original x. It might get multiplied by 2. Lambda could be 2. It might get multiplied by 0. Lambda there could even be 0. It might ge...

PROFESSOR: All right, now for something completely different. Before beginning though, I would like to raise a procedural question. It was my intent that quiz number two, which is like a little more than a week away, was going to be part symmetry and part tensors. I don't want to have your fortunes based by weight of 2:1 on symmetry theory as opposed to properties. What I would suggest, and you can express your opinion either way, is that we postpone quiz number two by maybe as much as 10 days, so that it can be half symmetry and half the introductory discussion of tensors. And if that does not create a conflict with some of your other classes, I would propose that as a suggestion. If not, we can just have on the quiz what we're going to cover of tensors in the next couple days and have the rest symmetry. So what I am suggesting is the quiz was scheduled for a week from November 8, which is a Tuesday. What I would suggest is putting that off to the 18th, and that ...