4

Let F (y + 2)i + (22 + 22)j + yk and S is the half-cylinder y = 0 and y = 13 Use Stokes' Theorem to evaluate1x2 betweenM x F) . ndo...

Question

Let F (y + 2)i + (22 + 22)j + yk and S is the half-cylinder y = 0 and y = 13 Use Stokes' Theorem to evaluate1x2 betweenM x F) . ndo

Let F (y + 2)i + (22 + 22)j + yk and S is the half-cylinder y = 0 and y = 13 Use Stokes' Theorem to evaluate 1 x2 between M x F) . ndo



Answers

Use Stokes' Theorem to evaluate $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$ $$ \begin{array}{l}{\mathbf{F}(x, y, z)=-3 y^{2} \mathbf{i}+4 z \mathbf{j}+6 x \mathbf{k} ; C \text { is the triangle in the }} \\ {\text { plane } z=\frac{1}{2} y \text { with vertices }(2,0,0),(0,2,1), \text { and }(0,0,0)} \\ {\text { with a counterclockwise orientation looking down the pos- }} \\ {\text { itive } z \text { -axis. }}\end{array} $$

Okay, let's go ahead and solve this problem. So we're given the vector F as X plus y squared called a white plus Z squared coma Z plus X squared. OK, and we are supposed to evaluate a line integral c f dot d. R. Where c is the curve that's drawn right here. It's an equilateral triangle with the Vergis is at 100010001 mhm. So if we try to evaluate this normally, which is something doesn't seem like it's too complicated to do. But the steps are going to be long because the parametric ization of C we can split it into three lines. So we basically have to evaluate three integral minimum in order to solve this problem straightforwardly. But we can use Stokes Terram in order to simplify the process a little bit. So let's talk about mhm. So instead of considering see as three lines a curve of three lines, I can consider it as an equilateral triangle as the surface s okay and because I know that the plane that contains this these three lines is the form X plus y plus Z is equal to one from there, we know that the normal, um, the normal vector is going to have the form at one comma, one comma one. And because when we dio ah, unit normal, when we use a unit normal vector to do the calculation, we just need to calculate the the magnitude of one comma, one common one which is equal to square root of three. So we can actually easily calculate that the unit normal vector is 1/3 off one comma, one comma one. Okay, so we're going to use that in order to solve the problem using Stokes there. Um, and let's recall that this is simply the double integral through the surface s off curl, yes dot ds, but I will use dogs in. He s okay. This is not the vector is just a surface in this particular case. So we would have to calculate what the curl of f is going to be. And that part, the calculation is not that difficult. So I will let you do this part, but it is basically we calculated like this. I j k. It's an operation to take the derivative partial derivatives with respect to X y Z and then we take the cross products. So this is going to be X plus y squared y plus Z squared and Z close X squared. I already wrote it down over here after my work. It simplifies very nicely. It'll be negative too, of Z comma X comma y Okay, so when we do curl s don't n what really happens here is that you will get negative to over square root of three and then you will have Z plus X plus. Why? But what we know about the plane here is that although we have it in different order, X plus y plus Z is equal toe one. So this simply goes away. So you will have negative to over squared of three which we can pull outside the double integral. Okay, so what we're going to evaluate is essentially negative to over three the double integral through S. D s. And then I like to use this technique all the time, but we know that this is simply the area off the surface. Okay, Now, this is an equal Atal triangle with side length, square root of two. So the area is Route 3/4 times Route two squared or, in other words, Route three over to. So when I multiply that number to negative two over Route three, you will get negative one, and that is the value that we're looking for.

Hello, everyone. Today we're going to solve the problem of Fight your functional face. That's where I that's look. The minus, like U K See, is a given but X squared. Plus why it square? Because what in the Exley, please? Or their Deco's zero? So closing the love half dot the year will be double integral or a sigma. They'll cross f dot in the so bill Cross f will be I. Nobody do. It's that square the don't by the way, two weeks. Okay, nobody does it minus like cute. So which comes to be like minus three was square I plus does that the plus toe kid so closed in Developer see her daughter They are Is it cool to double integral all the stigma minus tree? Why Square I Plus does that, plus two kids in the K delight the it's which is equals, doubling that'll over sigma to debate the X. So stigma is founded by X. Come away more or less off Hector Square. That's White Square. Let's Don Rickles what? Which is our commentator? Zero or less than recorded data less than miracles, toe by zero, less than record toe are less than recalls. Smart. So those Linda will oversee Earth door. The are will be zero toe by 0 to 1. The war the dictator which is to do are square 0 to 1, which is to put that standoff a question. Thank you.

Hello. I'm going to start problem number 17. She'll be hurt off. Validate. See, in the F dark Diya, which can be very introducing stocks tear up the given things are there because four minus into square minus y square aboard a a flight played and function 1/5 given by in the square in tow u to the power X minus way coma Well off my Squire preference and Zach Que Yes, he's that good. See, is the so I could with equation at this car plus life where requests for that is really good in the pain there because zero then exits choir plus life question Do less than or equal to four in the plane to their decor. Zero because let has been displayed SB there so far we don't this Okay, So he needs to find in order to avenge stocks here. A 1,000,000,000 curl of girls can be found by right. LF then the done end off I don't buy it. Elex and X coordinated X square into the about x minus right then day Don't buy it. All right In the room off Life choir plus life Okay in tow. Nobody does there Indo that cute resisted calls to I do. That is we will take this miners this job I know light and does a cube zero minus so bad over Does route over there is another toe again becomes a minus day in dude Dane, The bigger day Nolberto X off the Q panda, which is zero. So by those enough, it was quite a bit about X minus for just getting zero. That plus k indu les que don't know better X off route off like leverages. There are no extra minus so bad away off excess query pullup Rx minus like minus laser. Look up minus like again become my insulin. So they will cross f girl off That this girl will be Ah Okay, go into work which is in calls in the the rose their own. Okay, So in this regard yes will be going toe B A b s really equal to the so f dark. The are will be incognito self us integral left dark and the s, which is a photo surface And Deborah the year resisting calls area off s area off. Yes, ridges Because by our squad here Aliases to legal by India Four foot by that window. The question Thank you


Similar Solved Questions

5 answers
Ntne cla3ses datncned claasiner quaranteecRame Shadeneace Mabiz , the ooumalquacralic Claqonal neacnearest-nelcnbon.
ntne cla3ses datncned claasiner quaranteec Rame Shade neace Mabiz , the ooumal quacralic Claqonal neac nearest-nelcnbon....
4 answers
Tapping your toothbrush on the sink dries it off becausethe force of the tapping knocks it offthe vibration of the tapping lowers the friction between the water and the toothbrush_the water on the toothbrush has inertiathe heat generated by the tapping loosens the water
Tapping your toothbrush on the sink dries it off because the force of the tapping knocks it off the vibration of the tapping lowers the friction between the water and the toothbrush_ the water on the toothbrush has inertia the heat generated by the tapping loosens the water...
5 answers
Problems through deal with the RL circuit of Fig: 3.7.7, series circuit containing an inductor with an inductance of L henries, resistor with a resistance of R ohms, and a source of electromotive force (emf) but no capacitor: In this case Eq: (2) reduces to the linear first-order equationLI' + RI E()_SwitchFIGURE 3.7.7. The circuit for Problems through
Problems through deal with the RL circuit of Fig: 3.7.7, series circuit containing an inductor with an inductance of L henries, resistor with a resistance of R ohms, and a source of electromotive force (emf) but no capacitor: In this case Eq: (2) reduces to the linear first-order equation LI' +...
5 answers
Use the Laplace Transform to solve the initial value problem:V" + 4y' 12y "ev(o) =0, v (0) = 0.
Use the Laplace Transform to solve the initial value problem: V" + 4y' 12y "e v(o) =0, v (0) = 0....
5 answers
NdimRosac-icEt} 13.3.037.Hy NotesAsk Your TnacherHetAlualIl(w)iiWetticuiii splnEHAC]#n[30)1AalmprametrizationCcicn
ndim Rosac-icEt} 13.3.037. Hy Notes Ask Your Tnacher HetAlual Il(w)ii Wetticuiii spln EHAC] #n[30)1 Aalm prametrization Ccicn...
5 answers
2 Replace the following circuit with an equivalent circuit having only four gates. (Use only the NOT,; AND, and OR gates:)R
2 Replace the following circuit with an equivalent circuit having only four gates. (Use only the NOT,; AND, and OR gates:) R...
3 answers
Solve the given DE using Laplace transforms: Lx()} = F(p), Lx'()} = pF (p) - x(0) Lx" (t)} = pF(p) - px(0) - x(0)Sx'+ 4x =4, x(0) = 0, x(0) =2
Solve the given DE using Laplace transforms: Lx()} = F(p), Lx'()} = pF (p) - x(0) Lx" (t)} = pF(p) - px(0) - x(0) Sx'+ 4x =4, x(0) = 0, x(0) =2...
5 answers
Let $f^{prime prime}$ exist and be negative on the interval $[0,1]$. Show that if $P$ and $Q$ lie on the graph of $f$, then the line $P Q$ is below the graph, between $P$ and $Q$.
Let $f^{prime prime}$ exist and be negative on the interval $[0,1]$. Show that if $P$ and $Q$ lie on the graph of $f$, then the line $P Q$ is below the graph, between $P$ and $Q$....
5 answers
What component of the citric acid cycle can be produced from the carbon atoms of each of the following amino acids?a. leucineb. threoninec. cysteined. arginine
What component of the citric acid cycle can be produced from the carbon atoms of each of the following amino acids? a. leucine b. threonine c. cysteine d. arginine...
5 answers
Why is it better, when running a marathon, to drink a beverage with sugar for energy rather than one with amino acids?
Why is it better, when running a marathon, to drink a beverage with sugar for energy rather than one with amino acids?...
4 answers
Explain how horizontal and vertical ocean currents affect temperature.
Explain how horizontal and vertical ocean currents affect temperature....
5 answers
QuestionGiven the following reactions_ NzH4 02 N2 - 2 HzoAH =-632 kJHz - 02 H202AH = -278kJHz + 12 02H20AH = - 193 kJCalculate AH for the following reaction:NzHy + 2 H202Nz - 4 Hzo462 kJ462 kJc -SS2 kJ D.552 kJ 1103 kJ 0E
Question Given the following reactions_ NzH4 02 N2 - 2 Hzo AH =-632 kJ Hz - 02 H202 AH = -278kJ Hz + 12 02 H20 AH = - 193 kJ Calculate AH for the following reaction: NzHy + 2 H202 Nz - 4 Hzo 462 kJ 462 kJ c -SS2 kJ D.552 kJ 1103 kJ 0E...
1 answers
Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{n}{2^{n}} $$
Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{n}{2^{n}} $$...
5 answers
QUMSLAWANDRESISTANCENETWORKSAAA RESULISVoLTAGE AND CURRENT DATA FOR RESISTOR RyVoltage; 0,5 V)Current;Voltage, (+05W)Cumtent, [(15 mA)(15mA)9.549.$19.559-529.5105c9.539.512.019 $RESISTANCE VALUES FOR ExrERIMENT E7 Graphical Aeisunede (K2) (5)CalculatedI02 3153 4RsEkIs312.0RrARALALJ9.4
QUMSLAWANDRESISTANCENETWORKS AAA RESULIS VoLTAGE AND CURRENT DATA FOR RESISTOR Ry Voltage; 0,5 V) Current; Voltage, (+05W) Cumtent, [ (15 mA) (15mA) 9.5 49.$ 19.5 59-5 29.5 105 c9.5 39.5 12.0 19 $ RESISTANCE VALUES FOR ExrERIMENT E7 Graphical Aeisunede (K2) (5) Calculated I02 3 153 4 RsEkIs 312.0 Rr...
5 answers
04 Question (1 polnt)Ist atterptYcu are given the job developing : 2 CTub to give to anlmals that only targets bacteria If the drugonly targets structures that bxteria have the animal should rot experierce any side effects What structure is the = best target? Chooseone: cell wall plasma mcmbrare mitochcndria DNA
04 Question (1 polnt) Ist atterpt Ycu are given the job developing : 2 CTub to give to anlmals that only targets bacteria If the drugonly targets structures that bxteria have the animal should rot experierce any side effects What structure is the = best target? Chooseone: cell wall plasma mcmbrare m...
5 answers
Tnue or False: GTP Question 1Iruc OrFJISL: Question 2 1Truje orfalsc mulont31 thc cxtotoldonacing € cctrons
Tnue or False: GTP Question 1 Iruc OrFJISL: Question 2 1 Truje orfalsc mulont 3 1 thc cxtotol donacing € cctrons...
5 answers
Which of the following is NOT true about hot working?Select one: a. Deformation energy requirements are less. b It may be successively repeated because the metal remains soft and ductile Deformation is achieved at temperature above that at which recrystallization occurs. d. It produces an increase in strength with the attendant decrease in ductility because the metal strain hardens
Which of the following is NOT true about hot working? Select one: a. Deformation energy requirements are less. b It may be successively repeated because the metal remains soft and ductile Deformation is achieved at temperature above that at which recrystallization occurs. d. It produces an increase ...
5 answers
(a) Given the following matrices: -2 A = ( 3 7) , 8-( 4 c-6 ; He and D = Find, where possible: () A2 (ii) A? + D? (iii) € ' A (iv) A . € (v) € ' B [8 marks] (b) Calculate the inverse of the following matrix E, whereE= (55 -2:05 narics]Kel Calculate the eigenvalues of the following matrix:
(a) Given the following matrices: -2 A = ( 3 7) , 8-( 4 c-6 ; He and D = Find, where possible: () A2 (ii) A? + D? (iii) € ' A (iv) A . € (v) € ' B [8 marks] (b) Calculate the inverse of the following matrix E, where E= (55 -2: 05 narics] Kel Calculate the eigenvalues of t...

-- 0.019734--