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Hi! I'm Pritish Today, we will be talking about Trigonometry. This is the first video on Trigonometry. I would like to start with "Why do we want to study Trigonometry?" and "What is Trigonometry?" You would certainly know that Euclidean geometry is very fundamental to how we think about nature. The way we visualize things in Physics, for example, we use Euclidean geometry in the way we think. And often, Euclidean geometry requires some amount of innovative thinking to reason about problems. And often when you want to talk about Physics, you don't want geometry to be your bottleneck in thinking about Physics. So Trigonometry and Coordinate Geometry are two subjects which try to make a more systematic study of Geometry. And what is "Trigonometry"? Trigonometry is about a study of angles in an "algebraic" way. You would feel more comfortable doing Algebra Algebra is something you can do in a more mechanical way. Whereas, Geome...

POOJA REDDY: Hi. I'm Pooja. And I will be talking to you about how you can visualize the energy of defects, specifically focusing on screw dislocations. So crystals have an underlying lattice structure. But in reality, they're rarely perfect. They usually have mistakes in them which we call defects. A point defect can be a vacancy where an atom is missing from the lattice. And it can also be an interstitial, where there's an extra atom in the lattice. Another type of defect are dislocations, which are linear defects. Here is an example of an edge dislocation. The red arrow here shows a line direction in the defect, which is at the edge of this extra plane here that's in the lattice. Another type of dislocation is a screw dislocation. The blue arrow here shows a line direction of this defect. Here, items on one side are higher than the other. Here, it's lower. And you see this almost spiral shape that forms kind of like a screw. From these pictures you ...

STUDENT: Hello, everyone. Today, I'm going to talk about a very important model, analyze the energy band of a crystal, that is the tight binding model. Before I talk about tight binding model, let's now take a look at free electron model. The potential energy of free electron can be considered as zero, so the Hamiltonian of the free electron system is without the potential term. And the waveform function can be simply written as a plane wave. By substituting the wave function into Schrodinger equation, we can gather energy dispersion relation, which is a parabola in one dimension case. And we usually form the parabola into the first Brillouin zone. And this is the reduced Brillouin zone. And in this figure, we can see that the first band is at the bottom of the parabola. And the second band and third band, fourth, and so on. Similar to one dimension case, the energy of three dimension case is proportional to kx squared plus ky squared plus kz squared. For a given ...

SOPHIE CLYDE: Hi. I'm Sophie. PABLO VILLALOBOS: I'm Pablo. SOPHIE CLYDE: And this is our project, Night Hunter. So our early concept for this game was something like an invisible maze that you could only hear. So we had the idea of trying to-- the original idea came from the idea of superheroes who have enhanced senses. Since this was to be an audio-only game, we thought, how can we enhance your hearing? We thought, what if you could hear the things that you can't see? So the idea was that you would try to move through a maze. You would hear what-- you would use something like sonar to try to-- you'd send out a wave. It would make a sound. You could tell where it was in space and avoid it. That's pretty hard, as it turns out. One of the things we wanted to explore by making this game was how good at people-- or how good are people at actually figuring out where that invisible thing is. It turns out the answer is not very good. If you're trying to a...

PROFESSOR: OK. Finally, I'm going to solve this first order linear differential equation with a formula that works for any source term. So we've solved it for specific, nice, special source terms I'll remember later. But now we want a formula for the solution to that equation, period. And we want to understand the formula. So now, write the formula down, and then let's see why it's right. And then of course, we could put it into the equation and confirm that it's right. OK the formula is going to be-- this is the big formula you could say, for first order linear equations. So y of t. First we have the result of the amount-- I'm thinking of this as balance. The money in the bank is why it's increasing at this rate, because of interest being added. And it's increasing at this rate because of new deposits being added. And oh, maybe I should say about those deposits, I'm not thinking of like deposit once a year, or once a month, or once...

HiJohn Guttag. By day, I'm a professor of Electrical Engineering and Computer Science at MIT and also a member of the Computer Science and Artificial Intelligence Laboratory at MIT. But you probabaly know me as the figurehead or something behind 6.00 and the whole series of 6.00 courses. It's a really good question about what keeps a learner interested in pursuing this topic. We designed this course actually to be a hard course. It's very similar to what we teach at MIT. And we really didn't want to compromise on the opportunity for students to learn alot. However, as it's been said there's no royal road to mathematics. There's no royal road to computer science. To learn a lot, you actually have to work. On campus, we are able to provide the students who are struggling a lot of interaction and counseling and dare I say it hand-holding to encourage them to keep going. It's much harder to do that when we don't physically meet with the stu...

DAVID SHIROKOFF: Hi everyone. I'm Dave. Now today, I'd like to tackle a problem in orthogonal subspaces. So the problem we'd like to tackle: given a subspace S, and suppose S is spanned by two vectors, [1, 2, 2, 3] and [1, 3, 3, 2]. We have a question here which is to find a basis for S perp-- S perp is another subspace which is orthogonal to S. And then secondly, can every vector in R^4 be uniquely written in terms of S and S perp. So I'll let you think about this for now, and I'll come back in a minute. Hi everyone. Welcome back. OK, so why don't we tackle this problem? OK, so first off, what does it mean for a vector to be in S perp? Well, if I have a vector x, and S perp, and x is in S perp, what this means is x is going to be orthogonal to every vector in S. Now specifically, S is spanned by these two vectors. So it's sufficient that x be perpendicular to the two basis vectors in S. So specifically, I can take [1, 2, 2, 3] and dot it with...

DAVID SHIROKOFF: Hi everyone. I'm Dave. Now today, I'd like to tackle a problem in orthogonal subspaces. So the problem we'd like to tackle: given a subspace S, and suppose S is spanned by two vectors, [1, 2, 2, 3] and [1, 3, 3, 2]. We have a question here which is to find a basis for S perp-- S perp is another subspace which is orthogonal to S. And then secondly, can every vector in R^4 be uniquely written in terms of S and S perp. So I'll let you think about this for now, and I'll come back in a minute. Hi everyone. Welcome back. OK, so why don't we tackle this problem? OK, so first off, what does it mean for a vector to be in S perp? Well, if I have a vector x, and S perp, and x is in S perp, what this means is x is going to be orthogonal to every vector in S. Now specifically, S is spanned by these two vectors. So it's sufficient that x be perpendicular to the two basis vectors in S. So specifically, I can take [1, 2, 2, 3] and dot it with...

so uh I'm really pleased that we're able to have our speaker with us today John Goen Center John is the director of one of the science impact centers at Santa Gesa University and his Center is the indigenous Knowledge Center uh for Education hypen science impact it's part of the USGS science impact uh program and so John will present what for many of us is probably a different way of looking at the world so I won't take up any more time and and uh let John presentation you're not I'm not all right well U okay am I am I there okay well U personally I first of all I want to thank everybody for being here uh makes me feel very good even when I had the invitation I was like Wow gee you know I didn't realize what I had spoken about at the rest in Virginia already had a science impact but no I'm really honored really honored in fact uh speaking with some of my colleagues about the opportunity to uh go to another University and and present on beha...