Question

calc 3 help use stokes theorem to evaluate F*dR where F(x,y,z)= xyi+x^2j+z^2k and C is the intersection of paraboloid z=x^2+y^2 with plane z=y with counterclockwise orientation looking down the positive z axis.

Answer 1

Ok, so you're using stoke's theorem to convert a line integral to a surface integral...

Answer 2

I found curlF

Answer 3

so am i still solving for x?

Answer 4

Oh, I'm so stupid. I was going to evaluate the line integral. silly me.

Answer 5

lol, i know you're suppose to find curl F and R(u,v)

Answer 6

\[R(u,v)=u\mathbf{i}+v\mathbf{j}+\left ( u^2+v^2\right ) \mathbf{k}\]

Answer 7

can you check my curlF and r(uv)
curl = (0, 0, x)
and R(uv) = -2u^2cosv, -2u^2sinv, u)

Answer 8

and then i did the dot product between the two, and got u^2cosv

after changing the X in the curl to ucosv

Answer 9

Your curl is correct, but how did you do the parametrization of R?

Answer 10

no clue lol, I'm not completely sure on how to do parametrization

Answer 11

It looks wrong to me. If we take the x component squared plus the y component squared we get 4u^2, or 4 times the z component squared. However, we want that to equal the z component itself. Why not just use a simple parametrization like mine?

Answer 12

are you differentiating the x and y components?

Answer 13

Also, you need to take the surface differential \(\mathbf{R}_u \times \mathbf{R}_v\) and then you take the dot product of \(\mathbf{F} \cdot (\mathbf{R}_u \times \mathbf{R}_v)\) not \(\mathbf{F}\cdot \mathbf{R}\). I don't know why you are asking me about differentiating the x and y components, I did not do that.

Answer 14

\[\mathbf{R}_u\times\mathbf{R}_v=-2u\mathbf{i}-2v\mathbf{j}+\mathbf{k}\]If you use my parametrization. Then \(\mathrm{curl}(\mathbf{F})\cdot \mathbf{N}=u\)

Answer 15

but isnt the parameterization xi+yj+(x^2+y^2)k?

Answer 16

oh shoot, -2u^2cosv, -2u^2sinv, u

was suppose to be N

Answer 17

@funkeemonk3y Yes. That is the one I used.

Answer 18

k isnt the curl <0,0,x>?

Answer 19

@mickifree12 OH, sorry.

Answer 20

@Herp_Derp so when we change that to polar it's <0,0,ucosv>?

Answer 21

my bad, i wrote the wrong thing, so is my N correct?

Answer 22

so when we dot it isnt it just u^2cosv?

Answer 23

idk I was just doing this all in my head...

Answer 24

ok, so let's just say it was lol, time for the integration =[[

Answer 25

so when you have curlF * N = <0,0,x> * <-2u^2cosv, -2u^2sinv, u> right?