Okay, we're recording now. Okay, so what I'm going to do first is in R, So you have a function of one variable, f x, you know what its derivative is, right? And it was useful for many things. It was useful for studying graphs, it was studying for max-min problems and so on. Also useful for integration. Write partial derivatives came from the derivative. So that when you function of f, which depends on three or four variable, you can talk with partial derivatives. What we are going to study here are maps not just depend on and variables, but there'll be maps from our rent to RM. Okay, so I'm going to need to work in our ran, not just R2, R3. Now you didn't encountered three, you did R2 and R3. Everything you did in R2 and R3 works in Iran. Well, let me just quickly introduce some notation. Okay, so vectors, okay. So vectors, it's all RN set consists of vectors u, which are going to be written like this. And each u i is an element of art. Have you seen this notation here, this epsilon thing, right? It's INR is anyone who hasn't seen it. Ok. All it means it's belongs to. These are the set of real numbers. This means it just means belongs to. Okay? Just we use the same phrase again and again. So, okay. Now the vectors I am writing like this because it takes up less space, but, but you'll consider vectors u as a column vector. Okay? So that is, I'm, I'm always going to be thinking of them like this. So this is an n by one matrix. Ok? So in three dimensions, you had 11254 whatever. Okay? So this is an n dimension, but it take less space, my right, like this at times. Alright? Okay, so that's a vector. We can add vectors, right? So if you have u is u1 through UN, v1 through vn, then you'll get the obvious thing. You want to say, well does indeed new algebra, okay? So you can multiply a vector by a number. All right? Okay. There's a notion of length of a vector with length of u, right? It's obvious thing. Okay, you've seen this in your linear algebra, write the inner product space and someone said This was all. So this is the last thing is the dot product, right? So you have u dot v is U1, V1 plus. So if u is u1 through UN, alright? Okay. So what about the cross product? Is any cross product in four dimensions of two vectors? No. Okay, well, there is a dot product, right? What would you use cross products for in R3? What did we use cross products for? We're gonna have you found them useful on. They come up when you're trying to calculate something like the curl. Okay, curl of a vector field. Where else are you looking for? Just, just like within math or applications of it in other areas within, let's say. Okay. Don't you use it when you're trying to find the determinant of something. Okay, so determinant is used to find the cross product, right? Yeah. Ok. But notice how important the determinant was. Okay. So if you had just talking aloud, okay. Oops. Okay, time's up. But let me ask you, is could you come up with so if you wanted to use a determinant to come up with some, let me okay, let me keep promote. If you wanted to come up with a definition of cross product and you say, I'm going to use determinants. If you're in five dimensions. How many vectors you need to come up with the so-called cross product. If you're going to use determinant or for, right, you get the fold rows. And then that top row would be i j k or i j k, l, m. Right? So in higher dimensions it's not the cross product of two vectors. But it was something like if you're in five dimensional, you take the so-called cross product, therefore four-vectors, where you could come up with something but you couldn't do two, right? Two is very specific to three dimensions. All right? And you see the important role played by determinance. Determinants play a very important role in higher dimension calculus when you're doing integration determinant. Okay, anyway, so think about already in R3. Okay, that's enough. Let's stop. And thank you and we'll see you on Tuesday. Alright. So I'm going to stop my recording.
Vectors
From Rakesh Rakesh September 01, 2020
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