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Here you see a circuit powered by a battery and connected through a light bulb. This section of the circuit is made from copper foil. We can put a cut into the foil to make this circuit incomplete. Taking advantage of the thermal expansion properties of copper, we can place some candles under the foil to provide enough heat to expand the foil and complete the circuit again! Many materials, like metals, expand when you heat them. But other materials, like polymers, shrink when heated. The macroscopic properties of both of these materials are highly dependent on temperature. The difference in the macroscopic behavior of these two materials is determined by very different microscopic structure. Statistical Mechanics is the method used to describe and predict behavior at the macro-scale based on statistical models of microscopic behavior. The first step to understanding the power of Statistical Mechanics, is to use this method to define and understand temperature in terms of ...

Welcome to a new lecture on the binomial theorem. In this lecture, we will look at certain divisibility results which can be derived based on the binomial theorem. Once again, recall the binomial theorem is given by (x+y)^n = \sum{r=0}^{n}{C_r x^{n-r} y^r}, where we use C_r to denote nC_r. To give you a flavor of the kind of results we will be looking at today, let's look at the first example which asks you to show that 3^{2n+2} - 8n - 9 is divisible by 64 when n is a natural number. It asks you to show that this term is divisible by 64 for natural numbers n. At first sight, there doesn't seem to be a binomial expansion you can apply to generate this result so we would like to create a binomial expansion to derive this result. Notice that you have an exponent here, but you also have some terms here and these terms are, in general, not divisible by 64 for n being a natural number. You can plug in certain natural numbers and you can see that this term, which is bein...

Let's take a brief look at how to think about submitting your photographs as potential cover art in their presentations. The first thing that's essential is for you to study the journal layout to which you are submitting your image, how they might be placed in a particular cover format. Most covers these days are full bleeds. That is, the image takes up the whole cover and is therefore vertical. Some of them still have, let's say, rectangle places within the larger vertical format. But most of them are bleeds at this point. So you have to think about that when you do submit an idea for a cover. Think about where the journal's logo is placed, where additional text might be dropped. You have to take all these things into consideration to help the art director decide whether or not the image is good enough for a cover. But I do suggest not submitting a cover idea with the journal logo. Just let the folks at the journal use their imaginations. Let's start ...

Take a look at this music box for our first case study. So I place it on the glass of the scanner, and it looks pretty good. But as I look carefully. Which I hope you will do when you try this sort of stuff with your work, a couple of things that I'm not thrilled with. First of all, we're seeing some writing on the barrel. Looking closer, it appears to be numbers and letters. That's not great. Our eyes go straight to letters and numbers, and they basically attract our eye and actually become a distraction from the image, as far as I'm concerned. So I'm not happy with that. I'm also not thrilled with the positioning of the turning mechanism. Let's call it a stem. Somehow it doesn't feel composed properly. So what I simply did, was I played it a bit so the barrel no longer shows the writing, and I turned the stem so that it landed in a better composed image. I've been thinking about composition all my professional life, so see if you can ...

We're going to take a look at how point of view and composition are very clearly connected. But, before we show the connections, we have to sort of define, visually define, what I mean by point of view, for example. So, take a look at these four images. Now, it doesn't look it, but these images are of the identical sample, just simply taken with four different points of view. There, it's true that I also changed the background on them, but, generally, what I did was I moved the sample around or the camera, and we get four really completely different-looking visual ideas of what this stuff was about. Which one is right, I'm not sure there is a right in any of this. That's one of the fun things about this kind of work: There are no hard, fast rules that we have to follow. But I wanted, to give you, this example to show you that just by simply changing the angle or-of your camera, or moving your material around, you see different things. Again, we're ...

Suppose you get a text message. Your friend tells you to go to Lobby 7 at MIT to find the gift they left you 7 meters from the center of the lobby. Is that enough information to find the gift right away? As you can see, there are many locations 7 meters from the center of the room. Don't forget that we live in 3 dimensions, so there are actually even more points 7 meters away from the center of the room. Fortunately, in this problem, you can ignore most of them since we don't expect our gift to be hanging in mid air. Distance alone wasn't enough information. It would have been helpful to have both the distance and the direction. This video is part of the Representations video series. Information can be represented in words, through mathematical symbols, graphically, or in 3-D models. Representations are used to develop a deeper and more flexible understanding of objects, systems, and processes. Hi, my name is Dan Hastings and I am Dean of Undergraduate Educati...

GILBERT STRANG: OK. A third video about stability for second order, constant coefficient equations. But we'll move on to matrices here. So this is a rather special video. So this is our familiar equation. And I took a to b1, I just divided out a. No problem. So that's one second order equation. But we know how to convert it to two first order equations. And here they are. So this is two equations. That's a 2 by 2 matrix there. And so let me read the top equation. It says that dy dt is 0y plus 1dy dt. So that equation is a triviality. dy dt equals dy dt. The second equation is the real one. The derivative of y prime is y double prime. So this is second derivative here, equals minus cy and minus b y prime. And that's my equation y double prime, when I bring the minus cy over as plus cy, and I bring the minus b y prime over as plus b y prime. I have my equation. So that equation is the same as that one. It's just written with a vector unknown. It's a ...

PROFESSOR: So as a more technical aside, let's analyze more carefully the problem of release of viral load from a drop by process of diffusion. So here, again, I sketch a droplet, which would typically be an aerosol droplet in the size range of, let's say, microns. And the virion of interest has a size that is much smaller than that on the order of, let's say, 100 nanometers. And this white path is showing how such a virion would go from its initial position R, let's say, as a radio position, to the boundary. Now, the general problem of finding the expected first passage time from the point inside a domain to a boundary is a classical problem in the theory of stochastic processes and random walks. And it has the following representation. So the mean are expected first passage time from a point to an absorbing surface solves the following problem. Is the Laplacian of that time with a minus sign is 1 over D where D is the diffusivity. And so this is the DV t...

GILBERT STRANG: This is a topic I think is interesting. I like this one. It's about stability or instability of a steady state. So let me show you the differential equation. It could be linear, but might be non linear. dy dt is f of y. I'm going to-- I keep it that right hand side not depending on t, so just a function of y. And when do I have a steady state? There's a steady state when the derivative is 0. So if the derivative is 0 when f of y equals 0, let me call those special y's by a capital letter. So capital Y is a number, a starting value, where the right hand side of the equation is 0. And if the right hand side of the equation is 0, the left side of the equation is 0, and dy dt is 0, and we don't go anywhere. So the solution-- if f or y is equal to 0, then we have y stays at y. It's a constant for all time, and my question is, if we start near capital Y, do we approach capital Y as time goes on? It's, in that case, I would say, stable...

PROFESSOR: We've talked a lot about de Broglie saying that the wavelength is given by h over p. But we have not said much yet about the frequency of the waves. So what is the frequency of those matter waves? So what is the frequency-- frequency-- of the matter waves. So de Broglie did answer that same question. And the answer was obtained by analogy. We have p equal h bar k. And he said, well, just like the wavelength is determined by the momentum, we'll have e equal h bar omega. So the frequency-- so this equation is the one that now completes the story. Omega is equal to e over h bar. Fixes omega in terms of the energy. And we're going to say a few things. In fact, this will be an interesting digression into an important subject about waves that illustrates why this answer makes a lot of sense. And that's, really, all you can do at this moment. This is a postulate of quantum mechanics. That you do this thing, and with this, you get quantum mechanics. So ...

Any time you begin a communication, whether face-to-face, in writing, or over the phone, it's important to think about why you've opened that line of communication, what the goal of your interaction is, and how you're going to deliver your message. In this video, we'll watch 3 segments of a presentation created by MIT students. You will analyze the strategy used to develop the message, and hear how and why the presenters chose this strategy. This video is part of the Communication video series. Successful professional communication begins with the ability to analyze situational variables and make strategic decisions. Hi, my name is Joanne Yates, and I'm a Sloan Distinguished Professor of Management and Deputy Dean in the Sloan School of Management at MIT. After watching this video, you should be able to use a strategic approach in order to communicate effectively. This means that you will be able to Analyze your audience, context and purpose Examine an...

When you put a cold drink on the kitchen counter the counter surface temperature will decrease. But, if the cold drink is removed, the counter will eventually return to room temperature. If instead we place a cup of tea on the counter, the counter temperature rises; but if we remove the cup of tea, the counter top eventually returns to room temperature. We say that the counter at room temperature is a stable equilibrium. In this video, we discuss the world from the perspective of equilibrium and stability, and in particular linear stability. This video is part of the Linearity video series. Many complex systems are modeled or approximated as linear because of the mathematical advantages. All the world is an initial value problem, and the matter merely state variables. However, and less poetically, there are alternative interpretations of physical, and indeed social, systems that can prove very enlightening. The purpose of this video is to introduce you to the framework of...