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PROFESSOR: Next is this phenomenon that when you have a wave packet and it moves it can change shape and get distorted. And that is a very nice phenomenon that takes place in general and causes technological complications. And it's conceptually interesting. So let's discuss it. So it's still wave packets. But now we have to go back and add some time to it. So shape changes. So we had a psi of x and t is equal to 1 over square root of 2 pi phi of k e to the ikx e to the minus i omega of kt. And what did we do with this to analyze how it propagates? We expanded omega of k as omega of k0, which, again, this quantity is centered and peaks around k0, plus k minus k0 times d omega dk at k0 plus 1/2 k minus k0 squared, the second omega, dk squared at k0. And it might seem that this goes on forever. And what did we do before? We looked at this thing and we did the integral with this term and ignored the next. And with this term, we discovered that the profile moves wi...

Today, and for the next two weeks, we are going to be studying what, for many engineers and a few scientists is the most popular method of solving any differential equation of the kind that they happen to be, and that is to use the popular machine called the Laplace transform. Now, you will get proficient in using it by the end of the two weeks. But, there is always a certain amount of mystery that hangs around it. People scratch their heads and can't figure out where it comes from. And, that bothers them a lot. In the past, I've usually promised to tell you, the students at the end of the two weeks, but I almost never have time. So, I'm going to break that glorious tradition and tell you up front at the beginning, where it comes from, and then talk very fast for the rest of the period. Okay, a good way of thinking of where the Laplace transform comes from, and a way which I think dispels some of its mystery is by thinking of power series. I think virtually al...

-- is the next group of topics in 6.033 call fault tolerance. And the goal here is to learn how to build reliable systems. An extreme case, or at least our ideal goal, is to try to build systems that will never fail. And what will find is that we really can't do that, but what we'll try to do is to build systems which maybe fail less often than if you built them without the principles that we're going to talk about. So the idea is how to build reliable systems. So in order to understand how to build reliable systems, we need to understand what makes systems unreliable. And that has to do with understanding what faults are. What problems occur in systems that cause systems to fail? And you've actually seen many examples of faults already. Informally, a fault is just some kind of a flaw or a mistake that causes a component or a module not to perform the way it's supposed to perform. And we'll formalize this notion a little bit today as we go along. S...

KRISTEN: So our next keynote speaker is Alex Lorman. He is an engineer, but he has also been a professional photographer. And he's going to tell you what's inside the camera, and all the physics behind it, and some cool things cool he's done with his own cameras. So I'm going to turn it off to Alex. Thank you. ALEX LORMAN: I promise to bore you all to death with physics. It will be great, I promise. So like Kristen mentioned, I currently work as an engineer. I build robotic boats right now, which is all kinds of fun. But I used to build cameras and take pictures in a former life. It's OK, you guys can all have two careers, too. This is what I do. I play with sunken ships, and now I build them. It's great. It's good times. So also for cameras, helicopters are really useful if you ever want to take cool pictures. You should go out and ask your parents for one. So this is actually a camera I built. And that shot-- the picture on the left-- and bef...