5

(5 Consider the two surfaces Mi and Mz defined by the parametrizations:q1 (t,0)(sinh & cos t, sinh 0 sin t,t) (0, 0,t) + sinh & (cos t, sin t, 0) _ (cosh t ...

Question

(5 Consider the two surfaces Mi and Mz defined by the parametrizations:q1 (t,0)(sinh & cos t, sinh 0 sin t,t) (0, 0,t) + sinh & (cos t, sin t, 0) _ (cosh t cos &, cosh t sin &,+) _92 (t,0)Show that 91 R x R - M is a one-to-one parametrization of a helicoid (see section 4.1 exercise 9). Show that 92 is a parametrization that is not one-to-one Show that Mz is rotationally symmetric (see section 4.1 exercise 4) and can also be described by the equation 22 +y? cosh? z _ Show further

(5 Consider the two surfaces Mi and Mz defined by the parametrizations: q1 (t,0) (sinh & cos t, sinh 0 sin t,t) (0, 0,t) + sinh & (cos t, sin t, 0) _ (cosh t cos &, cosh t sin &,+) _ 92 (t,0) Show that 91 R x R - M is a one-to-one parametrization of a helicoid (see section 4.1 exercise 9). Show that 92 is a parametrization that is not one-to-one Show that Mz is rotationally symmetric (see section 4.1 exercise 4) and can also be described by the equation 22 +y? cosh? z _ Show further that this equation defines surface: It is called the catenoid Define a map F Mi 5 Mz by Foq1 (t,0) = 92 (t,0). Show that this map is smooth; not one-toone. but locally diffeomorphism.



Answers

Show that if the point $(a, b, c)$ lies on the hyperbolic parabo-
loid $z=y^{2}-x^{2}$ , then the lines with parametric equations
$x=a+t, y=b+t, z=c+2(b-a) t$ and $x=a+t,$
$y=b-t, z=c-2(b+a) t$ both lie entirely on this parabo-
loid. CThis shows that the hyperbolic paraboloid is what is
called a ruled surface, that is, it can be generated by the
motion of a straight line. In fact, this exercise shows that
through each point on the hyperbolic paraboloid there are two
generating lines. The only other quadric surfaces that are ruled
surfaces are cylinders, cones, and hyperboloids of one sheet.)

So being given the point, ABC, with an equation of Z equals y squared minus X squared. We can plug in the point, um, and we get that C is equal to B squared minus a squared. So now we have the Z equals y squared minus X squared. That's how we end up getting the C equals B squared minus a squared. So now we want to plug in the Parametric equation into the equation of the hyperbolic parable oId and solve for C. So we have X equals a plus t. We have that y equals B plus T and Z equals C plus two times B minus 80 then plugging in. Now that we're solving for C, we end up getting that C plus to be a T is equal to Z, but we can now write Z as B plus T squared minus a plus T squared. Then C is going to C plus two B T minus 2 80 is going to equal B squared plus two B t plus T squared minus a squared minus to a T minus t squared. And then we can combine like terms, and this is going to end up giving us a nice C equals B squared minus a squared. Well, that looks familiar because we found it up here as well, then plugging in a second Parametric equation. Um, we have X equals a plus t y equals B minus T and see again equals negative or sorry Z equals C minus two times be this time plus a team. We multiply everything out. We set Z equal to those other values. So now we have is a C minus to be T minus 2 80 equals B squared minus to B T plus T squared minus a squared minus 2 80 minus t squared. We subtract those things over, and once again, we end up getting C equals B squared minus a squared. So since both Parametric equations are equal to each other, then we can say that both of them lie entirely on Z equal to y squared minus X squared.

In this. In this question, we're going to prove that the hyper bullet of one sheet is a No, we're going to show that the hyperbolic Parable Oid is called the ruled surface by proving that the two lines labeled one and two which are both as parametric equations lie on this. Hyperbole Lloyd on this Parable Lloyd. In order to do that. We just need to show that we just need to substitute this information for Z, Y and X into our with our parametric equations. And then we can just prove your quality from there. So here's how we start this. Let's first expand these two binomial. We're going to get b squared plus two Bt plus t squared minus a squared minus 2 80 minus t square. So the t squares will vanish. Well the t spreads will vanish. And I'm going to rewrite this as B squared plus B squared minus a squared Plus two BT minus to a T. Now we want to show that we have d minus a we want to have b minus a in the parentheses. So allow me to just factor out a two times T From both sides from this to be T -280. Furthermore, B squared minus a squared equals C. Because see is a point on the proble. Therefore we can conclude that C plus two times B minus a times T equals C plus two times b minus a times T. Hence we've proven the first equation. Now let's prove the second equation. Remember the the second equation we have, it's going to be c minus two times Yeah, B plus A times T equals we're going to have b squared b minus t. The whole thing squared. Okay, subtracted from a plus T the whole thing squared. Mhm. So in doing so we're going to get that we have b squared minus two Bt plus t squared minus a squared minus 2 80 plus minus t square. Yes. And as obviously the t squares will vanish with the same logic as before. I'm going to bring the b squared minus a squared together. Yeah, so I have b squared minus a squared minus to b t minus 2 80. And once again I would like to factor out a negative one. Yeah, we can once again I would like to factor at the following term. She I'd like to factor out a -2 times T or two times or a two times, see whichever one you want because what we want to end up with is a plus inside the parentheses. Oh, so we will factor out a negative two furthermore. Again, b squared minus a squid is c squared. So what we want to show is we're going to say that C -2 times B plus a times T equals c minus two times b plus a times T. Hence we've proven our solution and we have finished the problem.

This problem were given six equations for surfaces and we want to identify what kind of quadratic surfaces they are. So here I've written the six different forums of quadratic surfaces and kind of with general coefficients like a BNC. And so we'll start with the first one. So we're kind of familiarizing ourselves with the different quadratic surfaces. So we have six X squared plus three y squared plus four z squared is equal to 12. So first, what we want to dio is when we have a second order terms in X, Y and Z, we want to normalize the right hand side. So we'll divide through by 12. We have one is equal to X squared over two plus y squared over four plus Z squared over three. And so this looks like an ellipse oId with a equal to the square to to be equal to two and C equal to square to three. Next up, we have y squared minus X squared minus Z is equal to zero. So noticed that Z is a first order term. So we're gonna reorganize this with Z is the first element, and we're also gonna make its coefficient one sold, but we'll multiply through by negative one. So we have Z plus X squared minus y squared is equal to zero. This looks like a hyperbolic crab Lloyd with a equal toe one and be equal to one. And next problem. We have nine x squared plus y squared minus nine. Z squared is equal to nine. So again we have a second order terms on the left hand side and a constant on the right hand side. So let's divide through by that constant to normalize, it will have one is equal. Thio X squared plus y squared minus. So sorry y squared over nine minus c squared. And since there's one negative term on the left hand side, this is gonna be a hyper high purple Lloyd of one sheet. So then we have a is equal toe. One b is equal to three and C is equal to one. Next, we have four x squared plus y squared minus four. Z squared is equal to negative four. So we're gonna normalize the left hand side started the right hand side, and so we have one is equal to so negative X squared minus weiss Word over four plus z squared. We have to negative terms. Therefore, this is a hyper Boyd of two sheets with a equal to one. Be equal to two and C equal to one. Next we have to Z minus X squared minus y squared. Sorry. So minus X squared minus four. Why squared is equal to zero. So first, what we want to do is we want to normalize the coefficient. Is he still have Z minus X squared over two minus two. Y squared is equal to zero. We have a linear term and to negative terms. So this is gonna be and elliptical Prabal Lloyd with a equal to square tissue and be equal to square of one half. Or we could write that as, um or to to over two, if you like. Next, we have 12 z squared minus three x squared is equal toe four y squared. So here we have all second order terms. However, we have no constant on the right hand side. So this is an elliptical cone. We'll move everything to the left hand side and then we want to normalize the coefficient, etc. So we have Z squared. So we're gonna be dividing through by 12. So is the squared minus X squared over four minus y squared over three is equal to zero. So then a is equal to two and B is equal to score three and that completes this problem.

This problem were given six different surface equations. We want to determine what kind of quadratic surfaces they are. So here we have kind of the basic forms or the six different types of quadratic services. We'll start with the first problem. So we have a so we have Z is equal Thio X X squared over four plus y squared over nine. I want to determine what kind of service this is and then what the different unknown coefficients are. So we have a winners each term and then to So if we pull everything on to the left hand side since we have living yours, each here and we're gonna be looking at the elliptical Prabal oId terms because there's only the ones with a linear variable. But we have Z minus X squared over two squared minus y squared over three squared is equal to zero. And this looks like our elliptic traveling formula where a is equal to to start an equal to zero. So we have a is two and B is equal to three. And the second equation we have Z is equal to why squared over 25 minus X squared over one so here again, we have a linear terms E So we're gonna be looking at the elliptical terms. So are elliptical crab Lloyd or hyperbolic tabloid. So if you pull everything out of left hand side, we have Z plus X squared minus y squared over five squared is equal to zero. So if we look, that's gonna be a hyperbolic crab Lloyd where a is equal to one and B is equal to five. Next, we have X squared plus y squared minus. C squared is equal to 16. So here we all have second order terms for our three variables, and we can rewrite the right hand side. As so we're looking at second order terms. We want to make the left hand side of the right hand side equal to one or resin will normalize. It will have X squared over four squared plus y squared over four squared minus Z squared over four squared is equal to one. So if we check, that's gonna look like a hyperbolic one sheet. I'm sorry. Hyper void of one sheet where a is equal to B is equal to C is equal to four. We have four squared in the bottom of each fraction. All right, Next up we have X squared plus y squared minus Z squared is equal to zero. So and see, we had unequal to constant, but this won't have unequal a zebra. So this one is gonna be an elliptic cone because we can rewrite this as D squared minus X squared minus y squared is equal to zero. And that fits the form of the elliptic cone with a equal Toby equal. The one next we have four z is equal to X squared plus four y squared. So we have a linear Z term and we have second order x and Y terms to remain everything the left hand side. And then we're gonna make a coefficient of Z equal that one we'll divide through by four. So we'll have Z minus X squared over two squared minus y squared over one squared is equal to zero. So this is gonna be an elliptic crab Lloyd with a equal to two and be equal toe want. And lastly, we have Z squared minus X squared minus Y squared is equal to one. So here the right hand side is already normalized and we have all of our variables as quadratic or a second order terms. So this is since we have two negative signs. This is gonna be hyperba Lloyd of two sheets where a equals B equals C equals one, and that completes the problem.


Similar Solved Questions

5 answers
Of the series interval of 'convergence 'convergence and Find the radius of - 10. ()k+2) n2"
of the series interval of 'convergence 'convergence and Find the radius of - 10. ()k+2) n2"...
5 answers
The total energy for the vibrating string problem can be written as E = Kinetic Energy + Potential Energy 6" Ju _ dx Jo J4 dc . Consider the case where u(.I) satislies the Watt equation with the boundary COn ditions Ux(.t) = "(L.4) Show that E is constant time Calculate the energy in mode_ Show that the total energy is the SUI of the energies contained in each mode_
The total energy for the vibrating string problem can be written as E = Kinetic Energy + Potential Energy 6" Ju _ dx Jo J4 dc . Consider the case where u(.I) satislies the Watt equation with the boundary COn ditions Ux(.t) = "(L.4) Show that E is constant time Calculate the energy in mode_...
5 answers
Rablen 6) Two Ooninuoush ce zlL hranskrva-hon: W: tv) & () sahf # {skn xyu ~ tyu + {xVz o 2xy ~ye+ vl = 0 nar (xsh)= (,1). Fr#e Uale &8 eck #ahs frmalon anel il efevential mahix a (W))
Rablen 6) Two Ooninuoush ce zlL hranskrva-hon: W: tv) & () sahf # {skn xyu ~ tyu + {xVz o 2xy ~ye+ vl = 0 nar (xsh)= (,1). Fr#e Uale &8 eck #ahs frmalon anel il efevential mahix a (W))...
5 answers
~mw RNUACA {J2;ibx 8-0.0-12. 9}: Cntazine vavtbz il2 <24u2n7€ #9 €ue criluetic 534J2095, g21 770: 0UC094 14 'ctV%n#jui: fecpucc_l @o ' RAVtF)ral Gcii
~mw RNUACA {J2;ibx 8-0.0-12. 9}: Cntazine vavtbz il2 <24u2n7€ #9 €ue criluetic 534J2095, g21 770: 0UC094 14 'ct V%n #jui: fecpucc_l @o ' RAVtF)ral Gcii...
5 answers
Point When we derived the - thin lens equation [Eq: (23.61. ]what did wc use? (choose all that apply) An assumption that the radius of curvature of the lens is much less than the distances and Similar trlangles Aray dlagram
point When we derived the - thin lens equation [Eq: (23.61. ]what did wc use? (choose all that apply) An assumption that the radius of curvature of the lens is much less than the distances and Similar trlangles Aray dlagram...
5 answers
Problem 2: Calculate the accelerating potential that must be applied to proton beam to give it an effective wavelength of 0.005 nm:
Problem 2: Calculate the accelerating potential that must be applied to proton beam to give it an effective wavelength of 0.005 nm:...
5 answers
Utath nc: Irurge 2 4 Io+t OeantIRLndenFn [eLFutanFstn AaIund Tea nenml AndnttuerilnFli2 "Fulutaal IEntMentieu
Utath nc: Irurge 2 4 Io+t Oeant IRLndenFn [eLFutanFstn Aa Iund Tea nenml Andnttueriln Fli2 " Fulutaal IEntMentieu...
5 answers
Finc Ine almeDenerhoa KRyii+inAnecisn Mtlon Aknnt: &a ii4eM I=Geneut
Finc Ine alme Denerhoa KRyii+in Anecisn Mtlon Aknnt: &a ii4eM I= Geneut...
5 answers
How many miles did Ashley travel from timeto time t = 2?At time t = 3,is Ashley s speed increasing or decreasing? Give eason for your answer:Is there time for 0<+<4 at which Brandon' $ acceleration is equal to 2.5 miles per hour? Justify your answer.0j Is there time for 0<t<? at which Brandon' $ velocity is equal to Chloe's velocity? Justify your answer:
How many miles did Ashley travel from time to time t = 2? At time t = 3,is Ashley s speed increasing or decreasing? Give eason for your answer: Is there time for 0<+<4 at which Brandon' $ acceleration is equal to 2.5 miles per hour? Justify your answer. 0j Is there time for 0<t<? at...
1 answers
One of $\sin x, \cos x,$ and $\tan x$ is given. Find the other two if $x$ lies in the specified interval. $$\sin x=-\frac{1}{2}, \quad x \in\left[\pi, \frac{3 \pi}{2}\right]$$
one of $\sin x, \cos x,$ and $\tan x$ is given. Find the other two if $x$ lies in the specified interval. $$\sin x=-\frac{1}{2}, \quad x \in\left[\pi, \frac{3 \pi}{2}\right]$$...
5 answers
The boxplot below shows the distribution of the prices for a single persons main meal at twenty restaurants in Melbourne;Ijo ImRkc u $Using the boxplot; what pcrccntagc 0f restaurants have main meal prices of lcss than S1202Select one: 2580508066,670812020Next page
The boxplot below shows the distribution of the prices for a single persons main meal at twenty restaurants in Melbourne; Ijo Im Rkc u $ Using the boxplot; what pcrccntagc 0f restaurants have main meal prices of lcss than S1202 Select one: 2580 5080 66,6708 12020 Next page...
1 answers
Evaluate. If an expression is undefined, say so. $$\frac{4+2 \cdot 7}{3 \cdot 2-9}$$
Evaluate. If an expression is undefined, say so. $$\frac{4+2 \cdot 7}{3 \cdot 2-9}$$...
5 answers
Usa normal approximation {0 find Ihe Plobabllty 0f the indicated number 0l volors: In this ca5? Os5me thal 149 eligible dee poeo 18-24 de Gndonyeeee( Suppose Provious sludy showud thal ami0ng eligible vo"Crs agod 18-24 ,2253 , Ihomn votod Probabilly Inat lewer Lhan 38 voledTha probability Ihat tower than 38 of 149 eligible volers votod Round t0 Iour docimal piaces 35 noodod
Usa normal approximation {0 find Ihe Plobabllty 0f the indicated number 0l volors: In this ca5? Os5me thal 149 eligible dee poeo 18-24 de Gndonyeeee( Suppose Provious sludy showud thal ami0ng eligible vo"Crs agod 18-24 ,2253 , Ihomn votod Probabilly Inat lewer Lhan 38 voled Tha probability Ihat...
5 answers
A penicillin solution containing 500 units/ml has a half life of 10 days. when will the concentrafion be 100 units/ml? assume first order degradation.answer: 23.2 daysshow work
a penicillin solution containing 500 units/ml has a half life of 10 days. when will the concentrafion be 100 units/ml? assume first order degradation.answer: 23.2 daysshow work...
5 answers
The equaiionS1v: - 322 +42 +3.2 + 12 lunchonol the neighborhood of the pa nt where = = 2 ,y = 2plicth delnesFindai this pointGie the exact valu es -
The equaiion S1v: - 322 +42 +3.2 + 12 lunchonol the neighborhood of the pa nt where = = 2 ,y = 2 plicth delnes Find ai this point Gie the exact valu es -...
5 answers
The volume of air in a room is such that when the air enteringis 110 f, 7600 btu must be extracted if the air is to be cooled to68 f in passing through the room. What is the volume of theair?A - 41,633 cu ftB - 10,931 cu ftC - 762 cu ftD - 1594 cu ft
The volume of air in a room is such that when the air entering is 110 f, 7600 btu must be extracted if the air is to be cooled to 68 f in passing through the room. What is the volume of the air? A - 41,633 cu ft B - 10,931 cu ft C - 762 cu ft D - 1594 cu ft...
5 answers
>O4-M Fawl te anea Caundeol ko Y = 9-X 2 1~r Y = 0i
> O4-M Fawl te anea Caundeol ko Y = 9-X 2 1~r Y = 0i...
5 answers
[ 220- @ HA 3 H | W F 6 V i W H 8 825 0 E 1 #W E 3 1 1 8 3 10 { 00 [ [ 1 T 1 [ 1 L 6i 1
[ 220- @ HA 3 H | W F 6 V i W H 8 825 0 E 1 #W E 3 1 1 8 3 10 { 00 [ [ 1 T 1 [ 1 L 6i 1...

-- 0.060493--