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Use Stokes' Theorem to find the circulation of F =4yi +3zj +7xk around the triangle obtained by tracing out the path (6,0,0) to (6,0,5), to (6,5,5) back to (6,0,0). Circulation = ∫CF ⋅dr
If you want to go from line integral to surface integral, shouldn't you need the anticurl?
Nevermind, they gave the anticurl
I mean A such that \[ B = \nabla \times A \]
But apparently they gave it to you already.
do you know how I would set up the line integral. I know that I have to use the points to make an equation of the triangle but after that I'm not sure
So to use stokes theorem, you need to parametrize the surface.
They don't want you to set up the line integral, but I can help you do that if you want.
The surface is the triangle?
I want to be able to do it using stoke's theorem/how they want it
Here is stokes theorem: \[ \iint_S\mathbf{F}\cdot d\mathbf{S}=\int_{\partial S}\nabla \times \mathbf{F}\cdot d\mathbf{r} \]
Now, the only way to use stokes theorem here is to do the left (surface integral) side. If you did the right (line integral side) would you really be using stokes?
No. I guess I was making a mistake. So how would I parametrize the surface? After that would the surface integral require a normal?
I'm getting line integrals and surface and green's theorem andstoke's theorem confused since we are barely covering them.
You can easily find two parallel vectors for the triangle right?
No, you do it by finding a vector between two points.
Between two points. So would I have to use points not contained in the triangle?
yeah so we want the bounds on the yz plane.
okay so since I know the yz plane is parallel to the trianle would I be able to choose any 2 points there?
oh so it is a projection onto the yz plane
okay so does that tell me what the limits of integration for z are going to be or am I supposed to use it to parametrize the triangle?
You could parametrize it as \[ \Phi(u,v) = (6,u,v),\quad 0\le u \le 5,\quad u\le v \le 5 \]
Oh I see. Because the x value is constant, right? Then from there will I have to calculte the normal to dot it with the curl of F?
In this case \(\mathbf{F}\) doesn't need to be messed with, you only take the curl to do the line integral part.
So I just dot the parametrization of the triangle with the curl and then I just use the limits of integration for u and v?
How many times do I have to say you don't take the curl?
Sorry I keep looking up at the equation. Okay. What would be the next step in my process?
\[ \iint_S \mathbf{F}\cdot d\mathbf{S}= \iint_D\mathbf{F}\circ\Phi(u,v)\cdot \frac{\partial \Phi}{\partial u}\times \frac{\partial \Phi}{\partial v}dudv \]
\[ \mathbf{F}\circ \Phi(u,v) = \mathbf{F}(x(u,v),y(u,v), z(u,v)) \]
\(D\) corresponds to the bounds of \(u,v\).
okay so it is the vector field f in terms of u and v dotted with the normal of the parametrization?
Okay I'm going to work it out really quick.
Okay so when I did the normal I got <1,0,0> which, then dotted with <4u,3v,42> So I got 4u then I took the double integral \[\int\limits_{0}^{5}\int\limits_{u}^{5}4udv du\] that became \[\int\limits_{0}^{5}20u-4u^2du\] from there I got \[10u^2-4/3u^3\] and my answer was 250/3. Did I mess up somewhere because my answer seems incorrect? Was my set up correct?
You have to consider the orientation.
the orientation is counter is wingspanwise so that means the normal is supposed to be supposed to be negative, right, or the parametrization of the triangle?
Okay I change the normal to negative and then my answer just ended up being -250/3 but it is still wrong. Does it look like I did a mistake in setting it up?
Do you know what the answer is?
Okay I got it I decided to use the parametrization that you gave me. Then I took the gradient of it and dotted it with the curl I got. so it was <-3,-7,-4> dotted with the gradient of <6,-u,v> which gave me 7-4 which is 3. Then I used all the limits of integration you gave me and got the correct answer which is 75/2 . I'm not sure if this is another part of the formula but it worked. Thank you very much! Would you gladly help me with one more wuestion if I post it up?