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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 ...

Here are two people. They're both standing, but they're standing in two completely different ways. Which person would it be easier to push over? If you wanted to push this person [left] over, where would you apply a force? What about this [right] person? The answers to all these questions can be explained using the concept of torque. 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. I'm Sanjay Sarma. Professor of mechanical engineering at MIT, In this video, we'll be talking torque and balance. In order to understand these core concepts, you'll need a working knowledge of vectors and their uses. Specifically, you must be familiar with force, displacement, and torque. We will also assume that you know how to compute a cross product,...

GILBERT STRANG: OK well, here we're at the beginning. And that I think it's worth thinking about what we know. Calculus. Differential equations is the big application of calculus, so it's kind of interesting to see what part of calculus, what information and what ideas from calculus, actually get used in differential equations. And I'm going to show you what I see, and it's not everything by any means, it's some basic ideas, but not all the details you learned. So I'm not saying forget all those, but just focus on what matters. OK. So the calculus you need is my topic. And the first thing is, you really do need to know basic derivatives. The derivative of x to the n, the derivative of sine and cosine. Above all, the derivative of e to the x, which is e to the x. The derivative of e to the x is e to the x. That's the wonderful equation that is solved by e to the x. Dy dt equals y. We'll have to do more with that. And then the inverse fun...

PROFESSOR: OK. Here we are. We've got our data. We've managed to plot it. Now let's go about constructing a matrix which will enable us to fit a polynomial to this data. First of all, I need a matrix. Here is a way to construct a matrix. We put s equal to-- The function zeros is a function which produces a matrix with all zeros in it, and I want this matrix to have dimensions endpoints by endpoints. So this command should produce such a matrix, and lo and behold, it does. Let me just clean up this display a little bit. At the moment, it's a little bit annoying that I'm plotting out absolutely everything that I have. This won't be too much trouble when I only have six points, but it might be in a lot of trouble if I had a lot more points. Here's how to stop our MATLAB or Octave from printing out the results of a command. All you do is you put a semicolon at the end of the line, like that. Or, like that. If I make that change, so you hit and run ...

PROFESSOR: So here is of something funny. You might say, OK, what is simpler? A theory that is linear or a theory that is not linear? And the answer, of course, a linear theory is much simpler. General-- Maxwell's equations are linear. Einstein's theory of relativity is very nonlinear, very complicated. How about classical mechanics? Is classical mechanics linear or nonlinear? What do we think? Can't hear anyone. Linear, OK. You may think it's linear because it's supposed to be simple, but it's not. It's actually is very nonlinear. Newton could solve the two body problem but he couldn't solve the three body problem. Already with three bodies, you cannot superpose solutions that you get with two bodies. It's extraordinarily complicated, classical mechanics. Let me show you. If you have motion in one dimension, in 1D, you have the equation of motion, motion in one dimension, and there are potential V of x, that this time independent-- a p...

GERALD SCHNEIDER: And here in the front that asked me whether a bird's song would count as a fixed action pattern-- doesn't sound fixed. This is because they learn a lot. But you know that's true for every fixed action pattern, every instinctive behavior. There's always some learning involved. It's never fully formed. Walking is a fixed action pattern in us. But in fact, we do learn things. And it develops a little bit over time. Some fixed action patterns have a lot of learning. And some, very little. An action like sneezing would have almost none. But in fact, if I look at various people, even in the same family, sneezing, they'll all do it a little differently because they've learned. They've added some learning on top of it. So we have a few things left from last time. I want to go over a few things. Examples of competition among axons during development, where axons will compete for terminal space-- they're either competing for gro...

chiefly in mind here uh two are coming from last time uh I wanted to say just a little more about this neat theorem of Max flow Min cut this was a this was a do an example of Duality and a particularly nice one and uh um chapter 7 of the book has uh examples and and more discussion of that then um you were kind enough to get involved in my new uh interest which was well the specific question was uh when can I complete a matrix given some of its entries when can I finish it out to have rank one and I have some news so of course I'm eager to tell you about it and uh and then the I I realized that it might be a pleasure to talk about Game Theory as another uh two person this is twers game theory created by uh V noyman and uh uh it's another example of Duality uh but uh just coming from somewhere from such a different source that it wasn't recognized uh for a little while that that it fit the framework okay with Max flow Min cut I just have this uh uh example to m...