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Stokes' Theorem parameterization

  • #1
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Homework Statement


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Homework Equations



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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.
 

Answers and Replies

  • #2
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?
 
  • #3
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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?
It would equal Fxdx + Fydy + Fzdz, but wouldn't it be simplified to just F?
 
  • #4
vela
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What do you mean "simplified to just F"?
 
  • #5
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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?
 
  • #6
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.
 
  • #7
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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.
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
vela
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Sounds like a plan.
 
  • #9
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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?
 
  • #10
vela
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You don't. The problem is asking you to evaluate the integral using Stoke's theorem.
 
  • #11
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You don't. The problem is asking you to evaluate the integral using Stoke's theorem.
So is the answer just:
t=0 to t=2pi (-e-cos2t/2sint+2sin2t+e-sin2t/2cost+4cos2t)dt
 
  • #12
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.
 
  • #13
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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.
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
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.
 
  • #15
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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?
 
  • #16
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.
 
  • #17
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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?
 
  • #18
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.
 
  • #19
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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>?
 
  • #20
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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  • #21
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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>?
 
  • #22
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.
 
  • #23
vela
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Do you have a textbook with worked out examples?
 
  • #24
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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.
 
  • #25
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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...
 

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