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Stokes' Theorem
I'm trying to evaluate the line integral around the curve obtained by intersecting the cylinder x^2+z^2=1 with the plane y=3. The vector field is F=<3xz, e^(xz), 2xy> and the curve is oriented counterclockwise when viewed from the right.
 one year ago
 one year ago
Stokes' Theorem I'm trying to evaluate the line integral around the curve obtained by intersecting the cylinder x^2+z^2=1 with the plane y=3. The vector field is F=<3xz, e^(xz), 2xy> and the curve is oriented counterclockwise when viewed from the right.
 one year ago
 one year ago

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SBurchetteBest ResponseYou've already chosen the best response.0
I parameterized the surface needed to use stokes' thm as r(u,v)=<u*cos(v), 3, u*sin(v)> where 0 <=u<=1 and 0<=v<=2*pi. The curl of F and the cross product of the partials of r(u,v) are quite long, but I got an answer of 6*pi which is far from the answer in the solution key.
 one year ago

SBurchetteBest ResponseYou've already chosen the best response.0
For the line integral it is, but we are supposed to use Stokes' theorem
 one year ago

amistre64Best ResponseYou've already chosen the best response.0
i can never keep the names straight
 one year ago

amistre64Best ResponseYou've already chosen the best response.0
what is stokes thrm by chance?
 one year ago

slaaibakBest ResponseYou've already chosen the best response.1
The line integral in space is equal to the flux of the surface
 one year ago

SBurchetteBest ResponseYou've already chosen the best response.0
\[\int\limits_{C} F.dr = \int\limits \int\limits_{S} curl(F).dS\]
 one year ago

slaaibakBest ResponseYou've already chosen the best response.1
what was your normal vector? <0,y,0? ?
 one year ago

amistre64Best ResponseYou've already chosen the best response.0
oy, its way to late for me to be doing anything remotely sane with that :/
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
If the plane intersects the cylinder you know the cross section's shadow on the xy plane is a circle right? So you need only to integrate around the circle? And since it's a surface integral r if FIXED at r=1 not to be integrated over. So integrate over z and phi (the polar angle).
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
Especially with y=3, its not even slanted so it will DEFINITELY only give you a circle of constant radius.
 one year ago

SBurchetteBest ResponseYou've already chosen the best response.0
The formula I have been following states that flux integral over a paramterized surface is \[\int\limits \int\limits_{D} F(r).(r_u \times r_v) dA\] This, in effect, eliminates having to compute the normal directly.
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
And since you're only integrating with scalars you don't need to worry about integrating over a "position" vector which would NOT give you a circle if the plain was slanted. And you still compute the normal directly, you just don't normalize it.
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
And I guess it would be the xz plane not xy.
 one year ago

SBurchetteBest ResponseYou've already chosen the best response.0
Integrating along the curve would likely be more simple, but we are supposed to use Stokes' theorem instead of directly computing the line integral
 one year ago

slaaibakBest ResponseYou've already chosen the best response.1
but the normal is quite obvious though, isn't it? It just points in the ydirection. so <0,1,0> is fine
 one year ago

SBurchetteBest ResponseYou've already chosen the best response.0
Yes, I meant we don't compute the nomalized normal, the wording gets tricky ;)
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
And I misread, since you want to calculate the line integral you only need to integrate over phi. So: dw:1354845022728:dw So: \[\Phi(\phi)=( \cos(\phi),3,\sin(\phi))\] That is your parameterization for that circle. So \[0 \le \phi \le 2 \pi\] And then you have: \[\oint \vec{F} \cdot d \vec{l}=\int\limits_0^{2 \pi} \vec{F}(\vec{\Phi}(\phi)) \cdot \vec{\Phi}'(\phi)d \phi\]
 one year ago

slaaibakBest ResponseYou've already chosen the best response.1
wait, I did this a long while ago... but if you just use xyz coordinates, and calculate the curl of the vector field, then dot it with <0,1,0> and then use parametrizations, it's not really tedious at all? since you're only dealing with the j component?
 one year ago

SBurchetteBest ResponseYou've already chosen the best response.0
That's what we need if we were doing the line integral directly, but to use stokes thm, we have to make it into a flux integral. and @slaaibak I didn't have too much trouble actually doing it, but the solution was incorrect and I can;t figure out where I went wrong
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
A flux integral of the curl of F.
 one year ago

SBurchetteBest ResponseYou've already chosen the best response.0
I just realized that I was checking the wrong section of the manual... so the answer I got actually was correct. Thanks for the help guys, I apologize for my oversight,.
 one year ago
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