Question

[Calculus 3] Can anyone help me with the following line integral? Thanks!

Answer 1

Use Stoke's Theorem to evaluate \[\int\limits_{C}^{ } F dr\] where F is given by \[F = <e ^{-2x}-3yz, e ^{-4y}+3xz, e ^{-2z}>\] and C is the circle \[x^2 + y^2 = 9\] on the plane z = 2 having traversed counterclockwise orientation when viewed from above.

Answer 2

I thought to convert it to \[\int\limits_{S}^{ }curl (F) dS\]but my friend thought to use \[\int\limits_{D}^{ } F'(r(t)) r'(t) dA\]I'm not sure what approach to use.

Answer 3

The question explicitly asks you to use stokes theorem right ?

Answer 4

Right, but once I converted to curl(F)*dS, I wasn't sure what to do from there. Is it just a simple conversion to polar coordinates from that point? I still would have a vector though that I can't integrate.

Answer 5

Stokes theorem says that the closed loop line integral equals the flux through ANY surface that is bounded by the loop. so start by picking a simple surface

Answer 6

Stokes theorem :

\[\large \oint\limits_{C} \vec{F}.d\vec{r}~~=~~\iint\limits_{S}\text{curl}(F)\cdot d\vec{S}\]

where \(S\) is ANY surface bounded by \(C\)

Answer 7

So does it make sense that the region S is given by the enclosed region of the circle x^2 + y^2 = 9? In that case, I could quickly convert to polar coordinates, with r's bounds being from 0 to 3 and theta's bounds being from 0 to 2*pi.

Answer 8

Yes you may choose the surface as \(x^2+y^2\le 9\) in the plane \(z=2\)

next, you need to find its normal vector