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Exercise IQ :(LO2, Recommended time: ntuePrecise the nature and the characteristics of the following quadratic surface: 2r2 +42+24+42- 8r- 3 = 0_ US Exercise LL :(L...

Question

Exercise IQ :(LO2, Recommended time: ntuePrecise the nature and the characteristics of the following quadratic surface: 2r2 +42+24+42- 8r- 3 = 0_ US Exercise LL :(LOS, Recommended timne: 18 min ) Given [ = dr "[y774 dy d: a) Draw the domain of integration: b) Fiud the value of USIg spherical coordinates. 4 Exercise_12 (LOL , Recomnmended tie: 25 minLet (C) be the curve enclosing the domain D given by: D {(4y) eR?: r+y < 2r. -2 1}.Let # = (r.f+y) be a vector field defined on R? Draw (C)Fi

Exercise IQ :(LO2, Recommended time: ntue Precise the nature and the characteristics of the following quadratic surface: 2r2 +42+24+42- 8r- 3 = 0_ US Exercise LL :(LOS, Recommended timne: 18 min ) Given [ = dr "[y774 dy d: a) Draw the domain of integration: b) Fiud the value of USIg spherical coordinates. 4 Exercise_12 (LOL , Recomnmended tie: 25 min Let (C) be the curve enclosing the domain D given by: D {(4y) eR?: r+y < 2r. -2 1}. Let # = (r.f+y) be a vector field defined on R? Draw (C) Find the counterclockwise circulation of H along (C) using: direct calculation_



Answers

In Exercises $7-12,$ use the surface integral in Stokes' Theorem to calculate the circulation of the field $\mathbf{F}$ around the curve $C$ in the indicated direction.
\begin{equation}
\begin{array}{l}{\mathbf{F}=x^{2} \mathbf{i}+2 x \mathbf{j}+z^{2} \mathbf{k}} \\ {C : \text { The ellipse } 4 x^{2}+y^{2}=4 \text { in the } x y \text { -plane, counterclockwise }} \\ {\text { when viewed from above }}\end{array}
\end{equation}

Were given a field and a curve in the direction and were asked to use the surface integral in Stokes theorem to calculate the circulation of the field around this curve. In this direction field is F equals X squared I plus two X J plus Z squared K and the curves C is the Ellipse four X squared plus y squared equals four in the X Y plane and the direction is counterclockwise when viewed from above. So first of all, make some calculations. The curl of our field f is equal to Well, if you do this in your head, we have zero I plus zero j plus tu minus zero. Okay, which is just equal to two K. This is the curl of our force field and he normal to our plane. This is going to be a positive K since curve is oriented so that its counterclockwise, when viewed from above and therefore the curl of our function f dotted with our normal just going to be too. And the differential de sigma of the surface integral is dx dy y. Putting this all together by Stokes is the're, um the circulation of this field around this curve, which is the integral oversee of the field. Yeah, is equal to the double integral over the region are bounded by this curve of the curl of F thought it within, which is to be a and of course, recognize that this is two times the area of our region are which is an ellipse. And we see from our ellipses equation that it has a major axis with radio length two and a minor axis of radio length one and so, the area of the ellipses to pie. And so this is equal to two times two pi for a total of four pi. And this is our answer.

For this exercise, we are going to use the fact that the double integral of F dot ds is equal to the double integral of F dot and D s. So that's where n is the unit Normal vector on DS is the magnitude of our sub You cross r c v d a. So what we want to do is right DS was being ableto n d A which would be plus or minus our view across our Savi. Over are the magnitude of our city across our serve the yeah times are Cebu cross R B the magnitude of that d a And that's just gonna give us plus in my ass RCB across our c v d a. So with that, um, we'll find RCB across our servi. First, we know that our view V is equal to U plus v u minus V one plus two, you plus V So when we take our you, we end up getting 112 An RV will give us one negative 11 When we take the cross product of those, we end up getting three one negative too. Eso then the DS that we end up having is going to be negative. Three negative one to because we make a negative. So that's d A then. Since we are given f of x y z, we see that f of our U V is going toe end up giving us this right here. It will be one plus two you plus the time each of the use squared minus B squared negative 31 times one plus two you plus B times e to the youth squared minus B squared and then you squared minus V squared. So with all that in mind, we can, um, take the cross product or not the cross product the dot product of F N ds. And what will end up getting as a result is a two times you squared minus V squared. And then since use between your own tu and vous between your own one, we'll have our bounds of integration. Now we'll bring this to out here, and this is D u D V. Then we just take these intervals will have the first integral, and we'll be left with two times the integral from 01 of eight thirds minus two V squared Devi and then Once we evaluate that, it'll be two times eight thirds minus two thirds, which is six thirds eso that'll give us 12 3rd, which is just four. Four will be our final answer, and that represents the flux.

All right. Number 47 given f r r cubed across the sphere of radius. A second of the origin can easily find the Parametric. Okay, then the partials. The respect to U and V. Okay, Now we cross these two together. Yeah, and right. And given the magnitude of our is X squared plus Oscar plus Z squared look in our X y and Z and we get our is you go to a R A Q. They cute in our function f could be sign if you co sign v really squared on you sign v Very squared because I knew you for a square now setting up our integral f ah, and our normal vector and evaluating that integral the fire zone 45 The fuck's the vector field. It's four by

The problem even bothered us as f which is equal. Do it Just glad do it on your display. Our is given as Gostin So Science t and you know we are resting which is equal to the Are you doing as minus 70 to go d you know now we have this If Dr Dubai if they are d v d. So you do given as s aspire to X on get Squire and do my last society the boss the you know Did he no blind have this sequence too? It is minus sci fi costarring last fall Cost our duty This is given Ask my guess. Saudi costs are dd last four years cost fire anything But as we know from lots that costs are equal soon Cost too deep less when you are Do therefore we wouldn't in the years in solving this problem. Yeah, So this sequence in solving this problem will assume cost e equals J My last Santee didi But it's indigent on the limited changed Cuba's 1 to 1 Juris fire in the saving last four innocent. You too to buy one less cost to the over to day. No, this spot because you over 31 it gives us Geo last four over you and we have this else deep, less signed to deep over do interesting little room five which will give us toe into by ness half signed for by a luminous science Generally leading this is you know, this is also zero. Therefore, the disease will do four inches, dancer.


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