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5 marks) Let S be a closed orientable surface, and let P be some point on the surface $ Suppose you have two paths a and B in S such that:both a and 8 start and end...

Question

5 marks) Let S be a closed orientable surface, and let P be some point on the surface $ Suppose you have two paths a and B in S such that:both a and 8 start and end at P- does not cross itself anywhere, and neither does 8; a and 8 do not meet anywhere except the start /end point P the intersection of & and 8 at point P is tranverse, not tangential (see the diagram below)TransvecseTangentialProve that the surface S must be of the form Torus # U for some closed surface U

5 marks) Let S be a closed orientable surface, and let P be some point on the surface $ Suppose you have two paths a and B in S such that: both a and 8 start and end at P- does not cross itself anywhere, and neither does 8; a and 8 do not meet anywhere except the start /end point P the intersection of & and 8 at point P is tranverse, not tangential (see the diagram below) Transvecse Tangential Prove that the surface S must be of the form Torus # U for some closed surface U



Answers

Surface area of a torus a. Show that a torus with radii $R>r$ (see figure) may be described parametrically by $r(u, v)=\langle(R+r \cos u) \cos v,(R+r \cos u) \sin v, r \sin u\rangle$ for $0 \leq u \leq 2 \pi, 0 \leq v \leq 2 \pi.$ b. Show that the surface area of the torus is $4 \pi^{2} R r.$ (FIGURE CAN'T COPY)

This problem, we will find the area of a tourists. So we're giving a tourist T obtained by rotating the circle in the Y Z plane of Radius A, centered at the 0.0 B zero about the Z axis. This is sort of like our surfaces a revolution except for an hour rotating a circle in the function This Taurus is pictured and figure 24 along with the process of how it's created, and we're told to assume that B is greater than a, which is greater than zero. In part, A were asked to find integral expression for the area of this tourist T oh, use symmetry. So by symmetry, as I pointed out before the area of the surface obtained by rotating the upper part of the circle is half the area of the Taurus and we rotated. Graph is Z equals G f y equals. And then this is a circle, which is centered at in the Y Z Plane B zero and has a radius of a. So this is going to be the square root of the radius, a squared minus. And then why minus center the y coordinate B squared and we have that. Why lies between? Well, since the radius is only a and the centers that why equals B Y lies between B minus A and B plus A. Therefore, a crime of why is equal to the opposite of why minus B over the square root of a squared minus y minus B squared and we have at the square root of one, plus the prime of Y squared. Well, this is going to be the square root of one plus why minus B squared over a squared minus y minus B squared and combining terms, this simplifies to square root of a squared or a since a is positive over square roots of a squared minus. Why minus B squared. So, using the previous formula, we have that the area of our tourists T is going to be twice the area of this rotated surface of rotation, which is two pi times. The integral from y equals a sorry, why equals B minus A to why equals B plus A of this is since Why is positive? Why times the square root of one plus g prime of y squared and plugging In our calculations, this is four pi times the integral from B minus a to be plus a of Let's see, we get a times why over square root of a squared minus Why minus b squared de y. And so this is a formula for the surface area of our tourists. Then in part B were asked, You find an explicit formula for the area of the tourists in terms of A and B. So, in other words, you want to evaluate the integral from part A. Well. To do this makesem use institutions. So let's make the use substitution. U equals Why minus b over a, and then do you will be equal to one over a de y and so this integral becomes four pi times the integral from you have B minus a, which is negative one to U of B plus A, which is positive one of and then we have eight times why? Well, you can solve you for why we get that y is equal to a U plus B. So this is a squared U plus a be over and then in the denominator, we have a squared minus and then why minus B is equal to us This is square root of a squared minus a squared, You squared And we have that DuSable too one over Ada. So we have a do you and simplifying this is four pi times and then we have the integral from negative 12 positive one. I actually break this up 22 different integral of a squared. And then well, not a squared. So on the bottom we can factor out a squared Take the square to get a positive a and then divide So we just get a you over the square root of one minus you squared, do you? Plus integral from negative one toe. One of then the a B over a is B over the square root of one minus. You squared, Do you? I'm sorry. I made a mistake here. Forgot to include this extra a So this is still a square to you And this is a B. Now notice that in our first integral the instagram is an odd function. Therefore, this is a zero integral. Now we have that our second integral is actually an even function. So it's symmetric about the origin and it's gonna be equal to four pi times two times the integral from 0 to 1 of a B over the square root of one minus. You squared, do you? And we see that in the anti derivative this is ate pie times a b times the inverse sign of you from zero toe one and plugging in. We get a pie a b times the inverse sign of one which is pi over two minus the inverse sign of zero, which is zero, which simplifies to four pi squared a B And so this is the surface area of our tourists in terms of a

Hi there Today we're going to calculate the SEF Syria off a Taurus. Now a tourist is a doughnut. And to do that, we're going to use surface off revolution by revolving a circle about the ex access, which will trace out the shape of a tourist. Now, the equation of this circle is X squared. Plus y must say or squared equals R squared. But we need that. The radius is strictly less than a so that we get this gap leaving those with, ah, hole in the doughnut. So, um, we were used the surface area Formula two pi in scroll between your limits of f of X one plus the square of the first derivative plus one or square root D X. To do this, we need to find why in terms of X, so take X to the other side. So then we have y minus A or squared is R squared minus X squared. Then we have why equals a plus minus. The square root of R squared minus X squared on the plus and minus is important because they refer, um, to the top and bottom of a circle. And we need to reflect both round. Sorry. We need to rotate both round so that we get the entire tourists. So we need We're gonna call this F plus or minus of X because they're two different functions that we need to you both on. We'll have s a plus on s a minus on dumb. The total surface area that we want to calculate is s a plus plus s a violence. So the first derivative is via the chain rule. We keep our plus minus because the A will vanish. We bring down 1/2 because the square root is a binomial power of 1/2 of them will subtract one from the half and that we can minus 1/2. So we'll get one over the square root. Then we take the derivative of what's ever inside our brackets on the derivative of R squared minus X squared is minus to x Andi, this will give us minus plus X over R squared minus X squared square root. And then we have the square of this is just X squared is same for both r squared minus X squared. In fact, we need one plus this on the way we're going to add one in this case is we're going to add R squared minus X squared over R squared minus x squared. Then these X squares council we have r squared over R squared minus X squared. Now we're going to consider, um, the plus and minus at the same time. So plugging our information in we have the integral to power times this from minus our to our Then we have our function, which is a plus or minus R squared, minus x squared square root. Then we have the square root of r squared. Ever ask a minus X squared, which is this Ah, over R squared minus X squared square root T x. I'm going to simplify this. We're gonna take how a factor of our here, Andi also a factor of our here, which will lead to some simplifications furthermore, will notice that this is an even function. If we change x two minus X, then we have the same integral. So what we need to hear is we need to flip our limits, so don't flip. I limits were set the bottom limit to zero because it's even over a somatic domain. So we set the bottom 120 and we multiply by two Safe for pie political from zero to our of a plus minus Ah, one minus X over our squared square. Right. And then when we take out a factor of our this will cancel. So I just have one minus x ever are squared square root d x Now we have one minus something squared and so we're going to choose a sign you substitution. So it's that you, um not you. We are going to let Exeter are equal. Sign feta and then DX is ah cause feet er de feta Have you noticed one minus X ever R squared square root is one minus. Co sign that cancer sine squared phones, word feta square root on what minus sine squared is Kirsten squared Theater square root, which is just cause theater. So from this we can see that d x over costs theater Samos DX over one minus x over r squared square root which is equal to R D Peter. And that is this part. So now we're going to ignore all our limits. Then we have, um the plus and minus are equal to four pi integral from zero to, uh, a plus or minus. Ah, and then we contract cause theater here and then this just de theater. Now, this is when we use total surface area. This is s a plus. It s a minus. This'd is, And there is n are missing here because it's actually our defeat. Er so you can take out four pi r And then we have the integral of a plus our cause. Theater defeat er Plessy Integral of a minus our cars Feta de feta. And that is to integral of a defeated. So this equals eight pi r a The integral defeated. So this is eight pi r a. Peter. Now we need to go back two x So sign feet. Er is X ever are so feet er is the inverse sign of X over our and so the surface area is eight pi r a The inverse sign of X over our on we put our limits is between his own are the inverse sign of zero zero on the inverse sign off. One is pi over two. So we have eight pi r a. Then we have poverty minus zero pie ever to which is for pi squared are a on That's our final answer

Eso for this question. We could work with the, uh, Bullock ordinance, you and B localized at, um, for any angle of rotation. New book that access Ah, distance, capital R from the centre of the cross section of the tourists cases, this circle. Well, it's a streak of tourists. A little kids. It'll be a circle. Um, but for privatization waken C game looking at the top figure of the two provided, um, that one being within a plane passing through is that accents? Um, so we can see that our AI Capone for our vector will be equal to a capital r Um, plus X, uh, in terms of the local localize polar coordinates, which will be our coast on you. So we have our paws capital R plus lower case R coast. I knew for the eye component. Well, so do you know this, uh, I'm committed the Asian vector to distinguish it from the lower case. R sk It already is. No, we have to recall. This is, um, with local coordinates. Lower case. Aren't you, uh, within the plane? Passing proves that access Turning in top figure. Sort of to define this with Inspector, the ex ally axes We also have to take into account, uh, the angle new, um, that we rotated about this That axis in the X Y plane. We gotta find polar coordinates. Uh, we have co sign new for the x component of this, um, of this x coordinate within the plane localized to that plane. Um, So again, just to recap first, locally within a plane arbitrary plane has computers that access we at the Capitol are over. Case are times cause you to get an X quarter within that plane into the final with respect to the global X coordinate. Because I knew that's that's our component. You know, J component is the find. Similarly, except we don't have to, um, what was the white component? We will. It will be completely analogous. Except instead of projecting onto the global hex access project on Global, why access the same local vector s? So instead of taking my mother co sign of new multiplied by the sign of new uh, the same sprechen Ah, as for the I cornered direction s so for the same reason I've been in a low class is quote a minute within plain prosecutors that access we have projected onto the global X Y accedes in order to derive our prime minister Ization Back began the the long cold It's denoted X value. The image within the plane R plus capital are slow case our clothes on you. Uh, but that's defines the vector with and go the global X Y plane. I wish we then have to project onto girls blacks. My axes isn't respectively Khosa, uh, co sign inside of you. That's r j component. Don't cry. Okay? Component within me. Local awareness. Remember Trey planning passing through this and axes? Yeah. Um, we don't need to take into account capital R. I was only from the ex flying plane. We just take our local, uh, is that 1/4? Which is our times, The sign of you in this case. And also, uh, this this that intact co responsible global Zander. So we don't otherwise needed modified its We had a project under the rubble. Ex My absence. When? So again, we have our sign of you. The local sad. A CZ we do for standing polar coordinates. Are you short of plain orange s? So that's our side. You got 1/4. Um, but the local warrant in this case, of course, part of the global one. So we also have our son cake or touch in Indy, This corresponds to the premises. Asian factor given in party. Good. Of course we do. Have you arranged between zero and two pi to range along the entire circular cross section of the tourists and then new is raging between zero and two. By also print across the entire global ex wife clean and for the surface area, uh, we could compute. You could set us up as surface integral, but we can take advantage. Ah, simple geometry problem. To drive it without integration more quickly, without intubation s. So we can see personally that the's a conference of our circular cross section will be too high. Lower case lower and observing that are, tourists can be viewed as a cylinder with height equal to the, um, circumference along the center of the Torahs, simply ah, wrapped around so that it's two ends meet you way. Just use the formula for the area of the cylinder, the surface area of the side portions elder, nor in the caps, which is the conference times. Hi, it's We've got some companies already. Then the high end using the formula for the circumference this time the circle ah, which runs along the center of the Taurus. Passing toilets, cross section centers. That's a conference is to buy Capital R. So you put that in there for the height of our cylinder. If we cut the tourists along one of its cross sections and stretch it on three and then we can see this does indeed simplified to four pi squared capital are times over. Case are, which is the formula given in question. Finish this.

In part a rest to find the parameter ization of a tourists. So from the figure you see that point on the circle C is going to have coordinates. X Z equals capital are plus lower case R co sign you and Z is lower case r sine you. And since the Taurus is obtained by rotating the circle about the Z axis in space, it follows that a point x y z is on The tourists with parameter ization are of UV equals. Then we have r plus r cosign you co sign the why is our plus Arcos on you So this is an I followed by r plus r co sign you sign of the J followed by recall it Z was our sign of you So this is simply our sign view okay. And so this is the parameter ization of the tourists where you ranges from 0 to 2 Pi and V also ranges from zero to pi on part B were asked to find the surface area of the Taurus So from part a we have the privatization So it follows that the tangent Fekter are you? This is going to be the vector negative. Our sign You cosign v I minus our sign. You sign v j plus our co sign You okay? And we have that The tangent vector RV is the vector negative R plus r co sign you sign v I plus R plus r co sign you co signed the J and then zero k So it follows that the normal vector are you Cross RV is the determinant of the matrix I j Okay, negative. Our sign You cosign v negative Our sign you sign v and our cosign you and negative R plus r cosine you sign the followed by r plus r co sign you cosign v zero. Yeah, In evaluating the determinant contain that this is going to be equal to negative R plus r cosine you times are cosine v Kosan You I then minus r plus r cosine you times are sign v co sign you j and finally minus our sign. You r plus r cosine you okay? And therefore we have that the norm squared of the normal vector. This is going to be after some simplification. R plus r cosine you squared. You can factor this out from all the terms and then we have leftover r squared CoSine squared V CoSine squared you plus r squared sine squared Vico sine squared you plus r squared sine square to you. We see that from the first two terms of in fact, out an R squared coastline square to you and add the coast and squared Vian sine squared be together to get one and then can factor in r squared from the r squared, cosign squared u plus r squared sine squared You simply obtain r squared So this simplifies to r plus r cosign you squared times r squared and therefore it follows that the norm of the normal vector. Are you cross RV? This is simply going to be r plus r co sign you. Times are because we know that our must be greater than zero and so must our plus our co sign of you. Therefore, it follows that the area as a double integral is going to be integral from 0 to 2 pi integral from 0 to 2 pi of or normal vector the normal normal vectors. So this is gonna be our plus our co sign you are, and then I want to choose the order with which we integrate carefully. So notice that we have a function which is a function of you. So we want to integrate with respect to you first, so do you. And then with your respective Devi. So we get integral from 0 to 2 pi of S o B you have our times are and then times two pies We have two pi r r plus R squared and the anti derivative of co sign you Well, this is going to be sign you This is r squared sign you but this is going to be zero minus zero which is still zero. So we simply end up with two pi r r Devi and we can factor out this constant and we get two pi times two pi r r which is the same as four pi squared are are And this is the area of the tourists surface area


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