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01:34

Scott N.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

01:02

Anshu R.

0:00

Simon E.

Jsdfio K.

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Okay, so here's a good example. So we want to evaluate this circulation integral over the specter field. And the curve is this triangle with Vergis is 00 So let me just go ahead and sketch out with this triangle. Looks like so one of the vergis is is on the origin, and then one is out here to to and then one is 02 and we have a counterclockwise orientation. So here is our curve c. Okay. And now this is actually very doable. I mean, we could break this up into one to three line and the girls and just add them up, add up sort of the contribution along each one, and that would give us the total circulation, but it would require doing three into girls. Okay, but let's see what happens when we apply green serum. So here's our curve. See, notice that because this is a closed, simple curve, it actually encloses a region that we call our okay. And so here's 12 just to just a label. Great. Okay, So what does green steer? Um, say the circulation version. It says that Okay, I want to find this circulation and a girl. And you know, I could parameter rise the curve, etcetera, etcetera. So that's one big bonus. I'm not going to have to prompt. Arise the curve if I use screens there. So what? This is equal. Teoh is the double integral over the region of So the basically the curl are really the Z component of the curl of this vector field que x minus p. Why? And that's over area elements. Okay, but let's just see what this Inter grand is. So this is P and this is Q So cute X is gonna be two x y and P y is going to be X over y So what I really want is the double integral over this region of what? Let's see of two x y minus x over Y d a. In this region is just a triangular region. So you know I can fix a why between zero and two, right? This is in a girl from 0 to 2 of why and then where does X go x goes from a fixed? Why zero up to that? Why value and then to x y minus x over Y this is going to be dx dy y okay, And then we can evaluate this says the integral from 0 to 2 of Okay, this is going to become just X squared. Why? It looks like and then evaluated from Let's see, just at why basically So yeah, that's expert. Why? So that's just gonna be y cubed. It looks like. And then what is this going to be? Well, it's gonna be X squared over two evaluated. So why squared over two y Or in other words, why over to why, Over to de y. Okay, so let's see. What do we get? We have y to the fourth over. Four minus y squared over four, evaluated from 0 to 2. What is this? Let's see to the fourth of 16/4, which is going to be four and then 4/4 is one. So it looks like our final answer is three. And that's the same thing that we get. If we broke this up into three line, Integral is going around the boundary. But we're getting around that by just integrating over the curl or the rotation within. So the rotation inside is this integral. The line integral is the rotation along the boundary. They're the same. They're both equal to three

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