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Considcr thc following paths: (i) a : [0, 2w] _ C, defined by a(t) = e2it _ (ii) 8 : [0, Vr] ~ €, defined by B(t) = 4+2e-2it?_ (iii) Y [0, 2w] , defined by -l...

Question

Considcr thc following paths: (i) a : [0, 2w] _ C, defined by a(t) = e2it _ (ii) 8 : [0, Vr] ~ €, defined by B(t) = 4+2e-2it?_ (iii) Y [0, 2w] , defined by -l+eit if t € [0,T) , ~(t) 2 - 4i -4+4i + if t e [T, 2w].(a) Sketch cach of the paths above and check if it is a contour. Detcrminc whethcr cach contour has clockwise Or anticlockwise oricntation. (b) Computc thc length of cach path:

Considcr thc following paths: (i) a : [0, 2w] _ C, defined by a(t) = e2it _ (ii) 8 : [0, Vr] ~ €, defined by B(t) = 4+2e-2it?_ (iii) Y [0, 2w] , defined by -l+eit if t € [0,T) , ~(t) 2 - 4i -4+4i + if t e [T, 2w]. (a) Sketch cach of the paths above and check if it is a contour. Detcrminc whethcr cach contour has clockwise Or anticlockwise oricntation. (b) Computc thc length of cach path:



Answers

Determine whether there is a path in the directed graph in Exercise 16 beginning at the first vertex given and ending at the second vertex given.
$$
\begin{array}{llll}{\text { a) } a, b} & {\text { b) } b, a} & {\text { c) } b, b} \\ {\text { d) } a, e} & {\text { e) } b, d} & {\text { f) } c, d} \\ {\text { g) } d, d} & {\text { h) } e, a} & {\text { i) } e, c}\end{array}
$$

Hey, we want to find a flow of each of the following passed from 00224 We have two graphs here. The 1st 1 party should be he so t are tee from 0 to 2. Okay, so I r t b y square, which is or t square and two x y, which is 40 square cane. And that's one to our crime. So just into bro from Joe to or wow two t square T t. So it has four times, uh, two Q Hideous, 30 32. That's for partly. Okay, for party here. Why is eagle so Access square? Okay, so that's he. He's where? Where he found Terrible too. Yeah, we have fr t should be wise. Where? Which is teaching the force to x y two ex wives with the two t to the cube. And our crime should be one to keep. So yes, so you can grow from to teeth. The forced us for tea to the force myself. Five t to the fourth tt. So just just one times two, the switch is 32 again. Candace, create another puss. Okay. In my house, we can use our one from year old hero, too 12 to 0 and are too, from 0 to 24 from 20 to 24 So are one that is tea hero and tea from Georgia to on. That is to t to still keep so for axes. Why Square, which is zero and two ex wives also zero. So he either grow for our one is just zero for Part two. Why Square said the or T Square and two x y to be eight. Heat right and our prime should be sterile. Two. Okay, so that's the answer. Should be from year old until try tea from here, too. On, that's the 16 t key, which is eight. He square he's where, and he's used to. So Times four. That's also 32. So the military's servitude for all three parts

Hello again about a problem asking for work done by a few Aah! Over three different path in a B and C um, in a It's just one straight. When life segment going from one is there is there one to one pie over two and zero and B is two line segment together. First is going from what's everyone going to the origin and then from the origin to one pie over to zero and in trees. Actually, three part one part is going down there and then going up there and then going on a parabola. So do you really ask herself really want to do these, like in agro, separate and three cases? Ah, we just noted that in old tree cases here A, B and C, the beginning point iss 101 and the end point is one pie over to zero is the same. So if it's similar to Problem 29 if we can show that this big half is conservative and weaken, then find the potential function, then we just go ahead. Do the potential function of the n value minus the beginning. So let's see, uh, you see the MMPI of been read. We're going to check that Partial off P would respect it too. Why? Food? It's going to be a long one. Say, look at that. Uh, what is that look like? Is going to be excellent. Want e off x y a y z plus cosa? Why, Luckily, it is the same that partial and which was wrecked. Izzy. Now we also need to check partial em. What? Respected to Z. And we get that it's his wife. Okay, that is the same with partial p with respect to exercise. Then we also need to check for sure and were well respected ex that just the Eazy e y z and it is the same. A partial ember would respect it too. Why that mean f is conservative. You good. So you have a potential function little to no, we go haven't find out. Potential function little if so. Partial. Um partial f would respect of X is going to be your m and your m is e t Y z. Wish me in your, um your efforts when they look like X y z here, then you take your show would respect it too. Why we're doing that would respect it toe once they got X z u y z plus for sure g what respected too. Why? And we compared with the intermix mover for our intern X e e e y z. Okay, Clough, Uh, whatever that iss Zico site why therefore your partial g would respect Why is going to be easy coz I want and then diggy going to be we integrate it with Respected why? So that we're gonna be disease sign Why? Plus on h function of only a Z in it. And there we need to find the h Im doing Parr. Show em what respected to Z the first guy, Xidan that what respected Z Ah, the only way up here this term. So what Respect disease have got a excellent why e uh y z plus, uh, where's my G tree? That guy would respect disease. I got signed off Why and then plus h prime of the and then we compared it with our P term score for the B term. The P term is X y e y z plus side. Why? So that means h prime is zero between ancient just a constant All right so conclusion your little f going to be x e y What, and we're done because now the inter grow, which is the work done, which is after the are to be f of the beginning of sorry after the ending, which is one pie over to zero minus after the beginning, which is 101 and we're going to just look in thes number, not one e lot off zero temps. I pi over, too. We'll see whiteness one easier time would close one side of zero. Let's see. And that was just the sea will be kids away and just be zero big zero That's it. See you in the next row Come into this world for all of A B and C, you don't have to do it separately. All right, and


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