Open course ware

The question of how far we can go when we enhance, or adjust, or touch up an image in science is a critical one. You are all familiar, I'm sure, with the stunningly beautiful Hubble images published all over the world. But most of the world is not familiar with the fact that the colors we're seeing were artificially created, decided and implemented by humans. You can read a conversation I had with some of these researchers in American Scientist, in the resource section, look for this thumbnail. I think you'll find the article interesting, about the decisions the researchers made about coloring the detail of nebula. So in essence, the coloring, or enhancement, was mostly done for the purpose of communicating structure, and I would say also for helping bring attention to these amazing images. Now remember, these pictures of the universe, are representations, that is re- presentations, they are photographs of that universe, they are not the universe. All the phot...

SRINI DEVADAS: How many of you have attended a celebrity party? I haven't. Well, if you count MIT professors as being celebrities, I've been to a lot of them, but that's not the case, right? People don't really think of us as celebrities, which is fine with me. So this puzzle, I call it the best time to party. And-- I'm having a little trouble erasing the-- oh you're right. There's another one? All right. Well, I'm probably going to run out of this. But you know what? This time I'll use-- I don't think I need this super thick chalk. There's a slightly-- I'll use this one. So setup here-- with all of these things, there's always a little bit of setup-- is, you got a ticket to this party. There are going to be all these celebrities. And it's a timed ticket with some flexibility. You only get to stay for a certain amount of time before you have to leave and other people are going to come in and hobnob with these celebri...

How do you design, test, and build a small and completely edible boat that propels itself through your cocktail? This is a complicated problem, but it--and many other problems like it--can be solved systematically, by breaking the problem down into several key phases. In this video, we'll explore how two MIT graduate students created these Cocktail Cruisers. This video is part of the Problem Solving video series. Problem--solving skills, in combination with an understanding of the natural and human-made world, are critical to the design and optimization of systems and processes. Hi, my name is Lisa Burton and I am Nadia Cheng. We are graduate students in Peko Hosoi's lab in the Department of Mechanical Engineering at MIT. Today, we're going to tell you about a class project that we worked on that turned into a product called cocktail cruisers. After watching this video, you should be able to identify the steps of the problem solving process and recognize that ...

PROFESSOR: How do we state the issue of normalization? See, the spherical harmonics are functions of theta and phi. So it makes sense that you would integrate over theta and phi-- solid angle. The solid angle is the natural integration. And it's a helpful integration, because if you have solid angle integrals and then radial integrals, you will have integrated over all volume. So for this spherical harmonic, solid angle is the right variable. And you may remember, if you have solid angle, you have to integrate over theta and phi. Solid angle, you think of it as a radius of one. Here is sine theta. So what is solid angle? It's really the area on a sphere of radius one. The definition of solid angle is area over radius squared, the part of the solid angle that you have. If you're working with a sphere of radius one, it's the area element is the solid angle. So the area element in here would be, or the integral over solid angle-- this is solid angle, d omega-...

The real topic is how to solve inhomogeneous systems, but the subtext is what I wrote on the board. I think you will see that really thinking in terms of matrices makes certain things a lot easier than they would be otherwise. And I hope to give you a couple of examples of that today in connection with solving systems of inhomogeneous equations. Now, there is a little problem. We have to have a little bit of theory ahead of time before that, which I thought rather than interrupt the presentation as I try to talk about the inhomogeneous systems it would be better to put a little theory in the beginning. I think you will find it harmless. And about half of it you know already. The theory I am talking about is, in general, the theory of the systems x prime equal a x. I will just state it when n is equal to two. A two-by-two system likely you have had up until now. It is also true for end-by-end. It is just a little more tedious to write out and to give the definitions. Here ...

PROFESSOR: How do we think of the Born approximation? Well, we can imagine a diagrammatic expression when we use propagators. Propagators are things that take a signal and propagate it in some direction. So think of your scattering center here, and, of course, we were supposed to look far away but I don't have enough room on the blackboard. I'll take this point to be far away. And now what does it say, this equation? It says that the first Born approximation, you get that r, the incident wave. The incident wave has reached r. This is the incident wave of r. So I will represent this as a wave that reaches the point r. That's the first term on the right hand side, a line reaching r. Because a wave has come in that direction and reached r. On the other hand, this is a little different. What is the physical interpretation of this term? This is a wave that reached r prime. And this wave was shaken by the potential and created the source, as if the potential of r pr...

PROFESSOR: All right, so how do we solve this? This is a very interesting thing, and I think it points to all kinds of important things that people find useful in physics. So here it is, the way, maybe, we can think about it. Think variational method. What is the variational method? You write a wave function and you try to see what is the expectation value of the Hamiltonian and that wave function. You calculate it, and now you know that the ground state energy is below that number. Because for the real ground state expectation value of the Hamiltonian is the ground state energy. For an arbitrary state it's more than that. So the variational method says, OK, if you want to figure out what is a good wave function, compute the expectation value of your wave function in the Hamiltonian, and then you will see, you will get some energy. If you tinker with your wave function you can get to the right energy. So this is what we're going to do we're going to try to tak...