flux-across-curves-in-a-flow-field-consider-the-flow-field-mathbfflangle-y-xrangle-shown-in-the-figu

Question

Flux across curves in a flow field Consider the flow field $\mathbf{F}=\langle y, x\rangle$ shown in the figure.
a. Compute the outward flux across the quarter circle $C: \mathbf{r}(t)=\langle 2 \cos t, 2 \sin t\rangle,$ for $0 \leq t \leq \pi / 2$
b. Compute the outward flux across the quarter circle $C: \mathbf{r}(t)=\langle 2 \cos t, 2 \sin t\rangle,$ for $\pi / 2 \leq t \leq \pi$
c. Explain why the flux across the quarter circle in the third quadrant equals the flux computed in part (a).
d. Explain why the flux across the quarter circle in the fourth quadrant equals the flux computed in part (b).
e. What is the outward flux across the full circle?
(FIGURE CAN'T COPY)

Joseph Liao · Accepted Answer

Our Prime 66 given F. Is why comma X. Compute outward flux for R. F. G. Equal to two. Co sign T. two sci fi. Mhm. From zero two Pi over two. So gradient is defined. Bye partial respect to X. Partial with respect to Y partial or the spec to see. So for a R. T. is to co sign TO two. Sine of T. Our prime FT. Good negative too Sine of T. To co sign T. The integral. EFTa. And yes Go to integral from zero. Hi over to of to scientific comma negative two eco sci fi ah two co scientific comma negative two Sine T. D. T. That equals in a girl from zero to pi over two for scientology. Cose I inti Plus four. Sine T. Cose I inti T. T. Or in a girlfriend pie overseas in trig identities of sign to T. T. T. Take the integral. Got four. It&#x27;s negative. Co sign to T. Over two, Evaluated from Pi over 220. That gives us four. All right party new privatization of our to co sai inti to sign T. Our prime FT. To go negative too sci fi. Two eco sci fi integral would be integral From pi over two hi of to sign T comma negative to co sign T. Ah To co sai Inti Come on a -2. Sine of T G. T. Now again this is gonna equal to pi over 22 pi a sign T. Co. Sine T DT which is you go to four cents in the grove power teared up. I signed to T T. T. The integral simplifies two negative coastline to T. Over two evaluated however, to two pi Which is -4. So that&#x27;s where fox is negative for parsi. Since the vector field F. In the normal vector and the first of the quadrants are negative. The flux is the same. So the flux is identical flux. Ah I tend to call and first and third quadrant party since F. And the normal vector and the second quadrant Um is negative as is in the 4th quadrant. Their product is the same. So flux is identical in second and four step E. Our flux is the combined flux across each quadrant. So net Does he go to 4 -4 plus 4 -4 Which is equal to zero.

Vikash Ranjan · Answer

in the problem they force fear is given us if off it&#x27;s going and that is equal. New white moments No, the bar Eni&#x27;s the car keys after wouldn&#x27;t two costing insanity for all t 20 by a two arresting equal minus two sci fi on you casting. So this is indeed a dusty Yes, that is zero. Why can&#x27;t it off our rafting dot R p Diddy No fun. This is digression by marking Teoh Sandy, go Gostin not read to Costea do finding DT then what this become by 12 it is full scientist e costing glass Full scientist costume Did you softer from the simplification? We have this asked for days and you buy a sign today Didi Aunt being done values we have this equal to four minus costs to be about two by two. So it&#x27;s become minus toe into minus student It is equal to this is full This is the handsome for the way have be well In addition, by wanting to find no sandy to Costea two costing Teoh Santee dating therefore these I want 02 for a scientist costing plus full Sandy Costin Didi which is equal. I want to four sign duty DT, which is equal to four into minus costs to keep on. I want to find it is equal to minus two and ghetto and this is equal to minus pole, which is the incident for part B off the problem. Further, we have I see something, Maxie. The part off this is settled that life that quarter when songs he belongs to three. Given that Sandino equal sand by people in the room and science part equal sense from people one, this must be the same as a on December 4 part harmonies for therefore the answer for this park scene will also be full when we have further man in body The part off this circle that last in the warrant concerns equals belongs to treat. Given that sine equals zero sense for my equal signs for people this is his own must be the same as in part B So in Barnaby they have the answers as minus full. So the answer for party also be my for no further be her last. Which is he? Well, swing. There are world class across the full circle is staff flat across all full off experts in each quadrant, which gives us two minus two plus two minus two equals zero. Therefore, this is the answer. Do them.

Yuou Sun · Answer

the problem. Certainly we can use formula Never night here do I? Minus and the ex. So I&#x27;m sure Beauty to a call sign and the way I should be a call sign, right? And my and the axe and should be minus three. Why so minus three? A sigh minus three A sigh And the axe The axe which is minus a Hello sign er the x minus. A sign minus a sigh. You see, right around 0 to 2 pi. So that should be a square. Ah, to co sign squared minus Sorry square. Be too cause I to t minus another two because I intend to Yeah, and this part and this part of all heroes because didn&#x27;t grow from 0 to 2 pi. So the answer should be minus one over two times to pine, which is finest pie a square. Hey. Okay, I see his minus pies were not pi square. If I have to use another one, not just the same dip Roll from 0 to 2. Pi and wax, which is a call sign toe a cost I and the wire which is a because I square minus and d y and should be X minus y so a C minus. Hi. Bye. And the X the X, which is minus hi. Minus a sigh. Okay. And then they have a Ah, two way square. Yeah. So a square also I to six. Where those Yes, finest. Where key, Which is This is one of us, cause I ain&#x27;t he This should be one or two signs to t and this this finest one over two. While mine is caused by Intuit E and all this part this part and this far gone because you grow should be zero. So the total should be one over two times too high times a square still should be high a square.

Joseph Liao · Answer

hi from 64 on the outward flux of field F goes X come a wife from a zero cylinder X squared plus y squared a square So for the parametric we get a co sign you comma a sign you comma v so we can find her And which is the partial respect to you across the park Shows like to be which is a cosign You signed you comma zero and we can set up our integral from 0 to 2 pi girl from like a l l a co sign you a sign you called zero dot a co sign you a sign you comma zero Do you d v giving us four pi l a squared? Yeah, Now it&#x27;s from the outward fucks at the field So we have to find which is our over the magnitude of our rocky are to the peas Gonna be x squared plus y squared r p over to which just comes out to a to the p Therefore, our vector fields gonna be x over a p come a y over ap comma zero now setting a part integral. We go from negative l two l integral from 0 to 2 pi after N d a. Which is integral from l and I go from 0 to 2 pi of a co sign square to you over a p plus a sine squared you over a p do you devi and that comes down to four pi al over a 35 p minus one. See? Okay. So for part C with final values of P of the flux is finite. So when a approaches infinity we have the flux equals zero for p grand. The one when a bridge is a Finney We have flux approaching infinity for P lesson one. And then as l approaches infinity way have the flux l over aid the p minus one a is fixed at an Albertsons Vinny, there&#x27;s there&#x27;s no P that you could substitute where the flux is finite. So the flux is also going to get a penny

Sriram Soundarrajan · Answer

colored. Today you&#x27;re going to see our problem about 43 here. The function of physical harm by to express why and function D&#x27;s view. And by X minus four way, we need to find the circulation as the less flux circulation was we&#x27;ll find but is have dark. They are physical do ability Bay boys minus no FBI. Don&#x27;t lie. End of the year. So this is going by. Well, there are Martin made us weren&#x27;t into the best kind of stripping zero And 2nd 1 is fax. It is going by Rosener, have Dorgan the This is it, Gordo. Over, uh, Dolev by no X plus those you buy No, like into the year. So are bigger Doing no f bible, It&#x27;s It&#x27;s like do my mouth full Indo the bridge is a good over minus. Do the u here The limit off our is far more pleasant, are pleasant, very critical for and it is quite the full 20 minutes before they were deserve from minus two. We do by ask went by foot. But this by foursquare by four my enough by one square by foot. Really good on minor 15 by by who

Yuou Sun · Answer

Hey, uh, problem buddy, too. We have a square now, actually, post material and actually forced Opie over to why it goes to zero and white pie over too. So that&#x27;s to use this formula here to compare the circulation partial empire tracks, which is called science y on minus part vampire tracks. Pasha am Packer. Why, Right, Graham, about Hawaii is minus Linus, cause I ax minus called honey. Why write about Hawaii? So that should be acts from zero to Howard, our two. You lie. And just times power too, right? So that&#x27;s what the so times, Cy. Why times why in minus zero. And I&#x27;m smiling too. That should be pipe. So the separatist be pie about the flux. Can you use this woman? Grandpa drags plus special why and other acts which is general, impartial and part of why. So that&#x27;s what the miners acts, Cy. Why, right? And just you took Yeah. Do you like? So that&#x27;s what the Because I Why, um times Ah, one or two X square. Right. So this is the minus one. So the total should be minus age and high square

Problem

Looking ahead: Area from line integrals The area …

Problem 66 Hard Difficulty

Answer

A. 4
B. $$-4$$
C. identical
D. identical
E. 0

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William Briggs, Lyle Cochran, Bernard Gillett

Calculus for Scientists and Engineers: Early Transcendental

Heather Z.

Kayleah T.

Kristen K.

Samuel H.

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A. 4B. $$-4$$ C. identical D. identical E. 0

Calculus for Scientists and Engineers: Early Transcendental

Heather Z.

Kayleah T.

Kristen K.

Samuel H.

Vectors Intro

Vector Basics - Example 1

William Briggs, Lyle Cochran, Bernard Gillett Calculus for Scientists and Engineers: Early Transcendental

Heather Z.

Kayleah T.

Kristen K.

Samuel H.

A. 4
B. $$-4$$
C. identical
D. identical
E. 0

William Briggs, Lyle Cochran, Bernard Gillett

Calculus for Scientists and Engineers: Early Transcendental