4

(a) Show that if x(u,V) is a regular parametrisation ofa surface S such that E = G = f(u,v) and F = 0, then the Gaussian curvature isK = A(ln(f)) , 2 fwhere 4 = 82...

Question

(a) Show that if x(u,V) is a regular parametrisation ofa surface S such that E = G = f(u,v) and F = 0, then the Gaussian curvature isK = A(ln(f)) , 2 fwhere 4 = 82 + 02 denotes the Laplacian:(b) Compute the Gaussian curvature of the hyperbolic upper half-plane H: a sur- face with a regular parametrisation x: R x R+ ~t Hwith Elu,v) = G(u,0) = 2 for some a > 0,and F = 0

(a) Show that if x(u,V) is a regular parametrisation ofa surface S such that E = G = f(u,v) and F = 0, then the Gaussian curvature is K = A(ln(f)) , 2 f where 4 = 82 + 02 denotes the Laplacian: (b) Compute the Gaussian curvature of the hyperbolic upper half-plane H: a sur- face with a regular parametrisation x: R x R+ ~t Hwith Elu,v) = G(u,0) = 2 for some a > 0,and F = 0



Answers

Consider $\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+x y \mathbf{j}+(z+1) \mathbf{k} .$ Let $E$ be the solid enclosed by paraboloid $z=4-x^{2}-y^{2}$ and plane $z=0$ with normal vectors pointing outside $E$
Compute flux $\boldsymbol{F}$ across the boundary of $E$ using the divergence theorem.

So for this properly, you can see the divergences either the wine minus eat with the white, which is zero that means spices diversions. Here we can con cruel thes to integrate the Symphony Hotel zero.

The Spectra field. But he's, uh, to x y along the direction plus x long J direction. Mom, Why? Plus c along the key direction on the for issue and by see equals or when a sexy squared minus y squared. See you later, darling. Google zero These describes service I would like to check us that, uh, use looks, uh, young holds by chicken. Huh? Both sides of the statement. So, girl, that not part of a normal vector outward to surface is so, uh, like the shape of these people to the interrupt over the boundary off s. Well, if that you Yes. Oh, we're here. The service. Uh, look, something like the falling off you g X And why is, uh is basically Seagal's former success perseids like, go up. So you have the boundary in the explosion plane. So of honorees The circle circle of Life is for for she wasn't square. That's what exploded on the egg stroke. Me? Um, on. Uh, we'll be Let's start by computer. These So will the co work first go? He's gonna be equal with terminal. These metrics. I g Okay, then we have the partials. What? You've always been. Thanks. It was to see and then now the components of our trip to X Y only word I want light, Lissy. Well, so that this girl is equal to I. I am so ourselves. These respect, why is one minus special? Actually, respect through she is you. So I plus J dams. But about this because you feel bad A J class, Katy, Times my shovel. Huh? X with respect to X is one. But along these, with respect wise to x y I just to exploit to x cooks on, uh, we'll be unit normal. Uh, well, what normal? We can compute my misty X around the Hardisty y so those would be to dungeon vectors. You are not leaving the planet in space. That surface. So the ex is Ah, Well, the tips of these transformation transformation puts in tech school eggs. Why do why I'm c two? No. Ah, this quantity Four men are sexy square minus y squared. So that the broke because of cross product. Well, the normal reaction off the first part off the ex with the way which is my G. Okay, the determinant been here. We have partial of these transformations. Respect works. So one partial of these transformation the simple Bible respect. Due back zero of the third with respect to X is minus two eggs Marshal of the first component district. Why, I should have decided to combine this with the right one. Bye. Settle that service with a y so that these better will be pulled too. I am. So, uh, my fence minus two x minus, minus x so two x along the high direction J kinds. Ah, well, so these are minor. The minus my master. Why? So through comes to one jenny times. So why run along the caterer ation? Well, just one last key. So these are, uh are normal vector on the since Kei is pointing up, it's a positive. Well, it's all aboard. So you and, uh, like todo these. Okay? Also the unit normal. It's, uh, the zen. So they normally see people too. X I bust through. Why, J. Lasky, you know, by the normal this vector that it's quite w if I did like it strong. So these and sort of money really seem to go on girl your apartment and we'll be doing our These is that so two legs. Ah, to like them zero cake that plus one minus the wakes over the normal w So you get one. When we were doing this interval with this cool about, we would like to pass the central I want to When we drove over the X y playing. So we use these politization and so some of you have any trouble on the X Y plane and then the into usually this one over W Warren over. W you know I love you on the radio program transformation. You'll have these come here. But this is the same with the normal w that these two council from the be having the dean to roll. We're they exploit plane of that region. But ah, this region x y plane you saw? He said these covered you stole interior bad. He's a disc bring. You still are. You still said that? Uh, the sea turtle here would be by EMS to squirt. I'm spoke since you don't four by huh? Or by, uh so these side of angel, she will go four by No, No. Well, hopefully we can get the same but excited. So we're gonna decide you do about a decision off on these curve. So yeah, you are. This can be described through 2% off. Say, don't long B I component and do sign off. Sarah, don't get your parents or a d s sequel to minors to sign off. I was too crucial. J So here, this is the x component. Please reveal what should I, uh, the s probably yes. Will be equal to so do X y huh? To have to x y times minus to sign off, huh? No. Plus, now these cynical point X exclaims to go sign Oh, there now are acceptable to because I'm upset this part here be ableto two times to absorb core. You could say up there That's co sign squared. And then you'll eggs people, girl, because it off there why is he going to do some confidence? You have to square on being I'm not square for eggs. On what? Them school Santa Flora. Then sign off there. But saying over getups word because you have a way. So that, uh, these we'll be doing this needle of these from zero to buy. Do you fear? Because, well, he's gives you known ties for They're being still want to buy going all the way around so well, for decent would have 16. We're being drill angel. Hello. So you know, and its purpose Concern of feta Because you're going to buy So by an unusual situation this interval to go home Um sign Cube era mother by free on give you compute these beginning to buy on Syria once I know goodbyes. Zero sign of 00 So these part right, he's always no, uh What? We were left, reader. You see drool. So we'll have so in terms of Syria to buy off for co sign squared off era there. And for that in general for was saying square off Florida, you can use data entity that that is equal to one plus because I have who Jenna. And so So that all these first thing tomorrow will be so four times. One house too. Pam's meeting 01 to buy which this part comes out to be more bye. And now for decent drill, huh? Who have two times? Drill, of course. Sign off to Kira from zero to to buy if there are people too, huh? So the sea drill is going to sign Caserta. You came cereal on to buy. Which is what? Zero? Because sign up or by people. Zero Once I know it's you zero. So you know. So So we see that the these other side it's also people to fall by No so great. Both numbers are people too, or by on Stokes theory holds for this case Go.

S starts to toe stays that India sequence. So it isn't dio the signal? No you have. Then they can sound purses and a chance. So this is condition this did you minus eight? Santa's far far Steve Bless for costs. For me, beauty is even asked Four way have to prove this four I mean the right hand side also No ticking, right? E had So we have Dan this sequence Four dynasty My phone went misty This is Goss so And he belongs to you for on a side not student did you know about And he was one of the land y minus two x I don't find or so saying twist My for this is minus four minus t sign So my for my misty cause Sudden do dizzy balls I know the This is even asked minus for my last d sounds say minus one of water for my Nestea is this course it is cost site Hey, no women because my reply these two things and we have Don't say with the door five outside Dar fire even s so hold dynasty Call sign Dynasty Science I Huh? We have just you Don't you have full when you know one. My has totaled over four moments speed because sign full of honesty. Call side before dynasty inside. So this is the sign, Okay? No, this is giving us. People do buy. You don't before. Yeah, these are This is your four. Therefore, we have emergency on a cheese. So the thing is, food on their is out is for that. This is the answer.


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