Open course ware

STEPHEN CARPENTER: I have two activities that I've prepared. So interpretation, this idea of constructing meanings or developing meanings about works of art, or we can think of interpretation as someone is speaking one language and then we take that language and we are able to-- not only do we speak and understand this language, then we can take those ideas and those words and turn them into a different language that someone else speaks. We can have interpretation function in that way as well. So it's about taking ideas from one context into another or one mode of representation in communication into another. But in terms of art and art practice, art criticism and interpretation work together. So this image I think is useful for our purposes here. Imagine that this image was the poster for a movie, full length, feature length movie. And imagine you were responsible for writing the copy-- for writing the text for the voiceover of the movie trailer, but all you had ...

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. Two equations, the question of stability for two equations, stability around a critical point. OK. So the idea will be to linearize, to look very near that critical point, that point. But now we're in two dimensions. So that's a little more to do. So here's the general picture, and then here is an example. So here's the general setup. We have an equation for the changes in y. But z is involved. And we have an equation for the rate of change of z. But y is involved. So they're coupled together. It's that coupling that's going to be new. So what's a critical point? Critical point is when those right-hand sides are 0. Because then y and z are both constant. So they stay at that point. Wherever they are at this critical point is steady state. They stay steady. They stay steady. They stay at that constant value. So we want that to be 0. And we want this to be 0. We have two equations, f equals 0, and g equals 0, two equations...

two and the topics are applications in one dimension uh and uh this is what's new this is the key part that's new here so now we're dealing with the question of finding the matrices finding the equations where do they come from and this is the going to be the key to that so again we we will end up with one of these symmetric products an A and an a transpose it happens that the a goes on the right and the a transpose on the left but the point is one is the transpose of the other and uh the product is of course square and symmetric okay but now where what's the application and these are going to be uh the matrices that come up uh if I create the right application and the discussion in the text is in section 1.4 I there're going to be examples with masses and springs because that's the most basic problem of mechanics so I know that your interests go all over the field of Applied Mathematics and uh lots of it so mechanics uh are sort of the natural top...

two big themes in in in lectures that are coming uh well there's several lectures on Signal processing and in particular on wavelet developments so that's a half dozen lectures that that are sort of a unit in the course uh the other major theme is maybe the right word would be optimization how do you how do you identify and compute uh an Optimum uh of course in freshman calculus you have a function f ofx you take its derivative and set it to zero now uh we got to move up from there to functions of several variables so we got partial derivatives and I'll start with that today and then functions of functions so where the unknown that you're looking for is not just uh a point but a function a point in function space we could say so continuous variables and then and then constraints is optimization when there are uh constraints on the allowed on the admissible uh choices that that's a big subject and uh it's partly a world of its own optimization it...