# Stokes' Theorem

Given c(t) = [cos t, sin t, 2 + sin (t/2)] where t $$\epsilon$$ [0, 2pi] and F(x,y,z) = (2-y + x2, x + sin y, $$\sqrt{}$$z4+1) --- Find $$\int$$F.dS over c(0, 2pi).

I've no idea how to do this... any help would be awesome! Thanks!

## The Attempt at a Solution

tiny-tim
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I've no idea how to do this... any help would be awesome! Thanks!

Well, what does Stokes' Theorem say?

HallsofIvy
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Actually, I see no reason to use Stoke's theorem. It looks like it should be relatively easy to integrate
$$\int_c F\cdot dc$$
directly.

If c= [cos(t), sin(t), 2+ sin(t/2)], what is dc?

Stokes' Theorem says that $$\int$$F.dr inside a boundary C = $$\int$$curlF.dS over a surface S... but how can I use it here?

And is dc = (-sin t, cos t, cos (t/2)/2) ??

tiny-tim
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Stokes' Theorem says ... but how can I use it here?

What is curlF here? Curl F is 3 k. I'm having trouble figuring out dS... I know for $$\int$$curlF.dS I need a surface with the given boundary curve for dS... but I'm stuck there.

HallsofIvy
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The reason I suggested NOT using Stoke's theorem but integrating directly on the path is that when $\theta= 0$, $(cos(\theta), sin(\theta), 2+ sin(\theta/2))= (1, 0, 2)$ and when $\theta= 2\pi$, $(cos(\theta), sin(\theta), 2+ sin(\theta/2))= (1, 0, 1)$. This is NOT a closed path and so Stoke's theorem cannot be used.

For some reason, perhaps having just waked up, I was thinking "cosine" instead of "sine". Yes, they are the same point, yes, this is a closed path.

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But if the end points of the curve are the same, isn't that the definition of a closed curve?

tiny-tim
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hi calculus-stud! Curl F is 3 k. I'm having trouble figuring out dS... I know for $$\int$$curlF.dS I need a surface with the given boundary curve for dS... but I'm stuck there.

(I make it 2k … have you copied F wrong? )

think … what is the flux of a (0,0,3) field through any closed curve? (and yes, it looks closed to me too)

Yeah it should be F = (-2y + x2....) sorry. So, curl is 3k.

The flux is zero? So, we don't really need to integrate or Stokes' theorem here?

tiny-tim
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hi calculus-stud! (thanks for the pm)
calculus-stud said:
The flux is zero?
no

what would be the flux of a (0,0,3) field through, for example, a rectangle or an ellipse in the z = 0 plane? Hey, umm, I'm kinda lost here. Can you give some more clues?

Ok, does it have to do with the orientation of the curve? I'm really stuck here ... any hints?

tiny-tim
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what curve? this is the integral of a vector going through a surface

Oh right, sorry... but then what is the surface? I'm so lost... I thought you were asking for flux of the vector field through any closed curve...

tiny-tim
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what's the flux of (0,0,3) going through the horizontal rectangle (0,0,0) (0,2,0) (2,2,0) (2,0,0) ?

or through the horizontal circle, radius r centre (0,0,0) ?

Dick