Stokes' Theorem parameterization

1. Apr 23, 2016

reminiscent

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
I only know that they gave the parameterization of the circle: r(t) = <cost, sint, 2>.
My problem is, did they already give the curl of F in the line integral? I don't understand why dx, dy, and dz are separated like that.

2. Apr 23, 2016

vela

Staff Emeritus
If $\vec{F} = F_x\,\hat{i}+F_y\,\hat{j}+F_z\,\hat{k}$ and $d\vec{r} = dx\,\hat{i} + dy \,\hat{j} + dz\,\hat{k}$, what's $\vec{F}\cdot d\vec{r}$ equal to?

3. Apr 23, 2016

reminiscent

It would equal Fxdx + Fydy + Fzdz, but wouldn't it be simplified to just F?

4. Apr 23, 2016

vela

Staff Emeritus
What do you mean "simplified to just F"?

5. Apr 23, 2016

reminiscent

For example, (F/dx)(dx), so the dx would cancel out. That happens to each. Actually, wouldn't it be 3F?

6. Apr 23, 2016

vela

Staff Emeritus
$\vec{F}$ is a vector-valued function. $F_x$ is the x-component of $\vec{F}$, not the derivative of $F$ with respect to $x$. Besides, the expression $F/dx$ is meaningless.

7. Apr 23, 2016

reminiscent

Oh right, so do I just take the derivative of the parameterization which will give me values equal to dx, dy, dz and plug that into the original integral? Do I also plug in the parameterization into F?

8. Apr 23, 2016

vela

Staff Emeritus
Sounds like a plan.

9. Apr 23, 2016

reminiscent

How would you go about simplifying the integral though? Some parts are fine but how about e^(-cos^2(t)/2), for example?

10. Apr 23, 2016

vela

Staff Emeritus
You don't. The problem is asking you to evaluate the integral using Stoke's theorem.

11. Apr 24, 2016

reminiscent

t=0 to t=2pi (-e-cos2t/2sint+2sin2t+e-sin2t/2cost+4cos2t)dt

12. Apr 24, 2016

LCKurtz

No. That isn't using Stoke's theorem. The other side of Stoke's theorem is$$\iint_S \nabla \times \vec F \cdot d\vec S$$Do that.

13. Apr 24, 2016

reminiscent

So do I find the curl of F first, plug in the parameterization into the curl, then multiply it with the normal vector using dot product?
I'm having a hard time relating these things I think.

14. Apr 24, 2016

LCKurtz

Yes

Not sure what you mean by "the" parameterization. You are given the parametrization of a curve enclosing an area. You are going to need to parameterize that area and use that. You will need two parameters for a surface.

You will have to show us what you do before we can tell if you are doing it correctly.

15. Apr 24, 2016

reminiscent

So far, I found the curl to be <-x, -y, 2z+2>. Do I find the normal vector by using the circle's parameterization?

16. Apr 24, 2016

LCKurtz

Let's call that vector you got $\vec V = \langle -x,-y,2z+2\rangle$, which looks correct for the curl. So now you need to integrate $\vec V\cdot d\vec S$ over the described area up in the $z=2$ plane. Your next issue is to figure out $d\vec S$. How do you find the area vector? If you look at the geometry for this problem, it should be easy to figure out a unit normal vector and an area element. You can worry about the integral after you figure out the integrand.

17. Apr 24, 2016

reminiscent

I think what I am confused about is do you use F or the circle to find the unit normal vector?

18. Apr 24, 2016

LCKurtz

You are given a curve which encloses a surface. Surfaces have normal vectors. It doesn't have anything to do with the vector $\vec F$. I suggest you draw a picture of your surface.

19. Apr 24, 2016

reminiscent

Is the unit normal vector just <cost, sint, 2>?

20. Apr 25, 2016

LCKurtz

No. That isn't even a unit vector. It is a parametric equation of your circular boundary. Surely your text tells you how to calculate a normal vector to a surface. Or you could just draw a picture of the surface for this problem and look at it.

Last edited: Apr 25, 2016