# Stokes' Theorem parameterization

## 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.

vela
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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?

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

vela
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What do you mean "simplified to just F"?

What do you mean "simplified to just F"?
For example, (F/dx)(dx), so the dx would cancel out. That happens to each. Actually, wouldn't it be 3F?

vela
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##\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.

##\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.
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?

vela
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Sounds like a plan.

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

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

You don't. The problem is asking you to evaluate the integral using Stoke's theorem.
t=0 to t=2pi (-e-cos2t/2sint+2sin2t+e-sin2t/2cost+4cos2t)dt

LCKurtz
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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.

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.
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.

LCKurtz
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So do I find the curl of F first,

Yes

plug in the parameterization into the curl,

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.

then multiply it with the normal vector using dot product?
I'm having a hard time relating these things I think.

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

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

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

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.

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

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

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

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

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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.
Okay, I have two answers I am unsure about for the normal vector. What would you say about <0,0,r>?

LCKurtz
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Okay, I have two answers I am unsure about for the normal vector. What would you say about <0,0,r>?
I would be confused about the use of ##r## because you are using it for the curve so I'm not sure what you are thinking.

vela
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Do you have a textbook with worked out examples?

I would be confused about the use of ##r## because you are using it for the curve so I'm not sure what you are thinking.
How about <sint, -cost, 0>? Sorry, I came up with these with my classmates.
Do you have a textbook with worked out examples?
No, we just depend on MyMathLab and the examples our professor has done in class, but we have not done anything like this.

I would be confused about the use of ##r## because you are using it for the curve so I'm not sure what you are thinking.
He also did not really explain well how to find the unit normal vector...