Open course ware

DAVID JORDAN: Hello, and welcome back to recitation. Today the problem I'd like to work with you is about computing partial derivatives and the total differential. So we have a function z which is x squared plus y squared. So it depends on the two variables x and y. Now the variables x and y themselves depend on two auxiliary variables, u and v. So that's the setup that we have. So in part a, we just want to compute the total differential dz in terms of dx and dy. So u and v aren't going to enter into the picture. And then in part b, we're going to compute the partial derivative partial z partial u in two different ways. First, we're going to compute it using the chain rule. And then we're going to compute it using total differentials. And so we'll substitute in some of the work that we had in a to solve that part. So why don't you pause the video now and work on the problem. We'll check back and we'll do it together. Hi, and welcom...

DAVID THORBURN: One of the things I aim for in the course, is to try to show students, in ways as concrete and rich as I can, what the authority, or power, of movies in a cultural moment might be. And I try to give them some examples of that. In my first lecture, I sometimes read to them from a passage from James Agee's novel, A Death in the Family. In the very beginning of the novel he describes, beautifully, an excursion between a father and son, going to see a Chaplin comedy in 1915 in some southern city. And what he describes is what a social occasion it was, what a moment of bonding it was for father and son, what a cultural moment it was more broadly for the whole of the population that went into the theater to laugh uproariously at Charlie's shenanigans. "And then the screen was filled with a city, and with a sidewalk of a side street of a city, and a long line of palms. And there was Charlie. Everyone laughed the minute they saw him squattily walking,...

[MUSIC PLAYING] DAVID KAISER: Rockets, incredibly powerful clocks-- soon, lasers-- what hasn't been impacted by these things? SARAH HANSEN: Today on Chalk Radio, my guest is David Kaiser, a professor in the physics department at MIT, who also teaches the history of science. His course, Einstein, Oppenheimer, Feynman: Physics in the 20th Century was recently shared on MIT OpenCourseWare. Professor Kaiser is such an interesting person to get to interview. Simply thinking about how to interview someone responsible for teaching the history of, well, everything from the Big Bang to how hippies saved physics-- and, yep, that is the title of one of his books-- was humbling. There was so much here that we wanted to share with you, so we decided to air this interview in its near entirety. Professor Kaiser explains the earliest moments of the universe and how those moments help us understand our current night sky. We talk about how culture, politics, and even physical spaces sh...

DAVID THORBURN: The film experience is, as I said, began many years ago, 38 or 39 years ago in the version that I teach. And it has undergone almost a continuous process of transformation. One of the ironies is that it began its career in an era before there was any technology available to support film study. It made things actually very awkward. But it was also in some way convenient for the professor. What we used to have to do was show the film. We used to-- and in fact, MIT belong to a group called the University Film Study Center. A group of institutions, included Harvard and Yale. And what we did was we had a library of 16 millimeter prints of classic films. And we circulated them. So you had to alter your syllabus so that the guy at Harvard was also able to do Keaton because there was only one copy of The General available. So that was very complicated. And but we all managed to do it. And then at while I was at MIT, that system became moribund because videotape be...

DAVID JORDAN: Hello, and welcome back to recitation. So today, the problem I'd like to work with you is about taking partial derivatives in the presence of constraints. So this is a pretty subtle business. So take your time when you work these problems. So what we have is we have this function w, and it's a function of four variables: x, y, z, and t. OK? But it's not really a function of these four variables because we have a constraint. So we want to study how w changes as we vary the parameters, except that we have imposed this constraint here. So that really we kind of only have three variables, because we have four variables and one constraint. So that's what partial derivatives with constraints help us do. So let's explain first the notation. OK? So it says partial w partial z, and then we have the subscripts x and y. So what's important about this notation is not what you see as much as what you don't see. What you don't see is the va...

DAVID SHIROKOFF: Hi everyone. Welcome back. So today I'd like to tackle a problem on pseudoinverses. So given a matrix A, which is not square, so it's just 1 and 2. First, what is its pseudoinverse? So A plus I'm using to denote the pseudoinverse. Then secondly, compute A plus A and A A plus. And then thirdly, if x is in the null space of A, what is A plus A acting on x? And lastly, if x is in the column space of A transpose, what is A plus A*x? So I'll let you think about this problem for a bit, and I'll be back in a second. Hi everyone. Welcome back. OK, so let's take a look at this problem. Now first off, what is a pseudoinverse? Well, we define the pseudoinverse using the SVD. So in actuality, this is nothing new. Now, we note that because A is not square, the regular inverse of A doesn't necessarily exist. However, we do know that the SVD exists for every matrix A whether it's square or not. So how do we compute the SVD of a matrix? We...