d) Next evaluate the middle expression directly, the flux integral of the curl of F through the disc D. Remember to be careful with the orientation of the normal vector of D; it must match up with the counterclockwise orientation of C via the right-hand rule. This should happen "naturally" but double check anyway. e) Finally evaluate the rightmost expression directly, the flux integral of the curl of F through the cone E. Same warning about orientation applies here.
d) Next evaluate the middle expression directly, the flux integral of the curl of F through the disc D. Remember to be careful with the orientation of the normal vector of D; it must match up with the counterclockwise orientation of C via the right-hand rule. This should happen "naturally" but double check anyway. e) Finally evaluate the rightmost expression directly, the flux integral of the curl of F through the cone E. Same warning about orientation applies here.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Calculus III question. Need help with d) e)
Transcribed Image Text:Stokes's Theorem / The Curl Theorem
2. Consider the vector field
F(x, y, 2) = (yz?,x,x + y)
and the closed curve
C: r(t) = (cos t, sin t,0) 0sts 27
Note that C is the unit circle in the xy-plane traced out counterclockwise (as viewed from above).
Also let D be the unit disc in the xy-plane and let E be the 45-degree cone whose tip is at the point (-1,0,0) and whose
boundary is the curve C. That is,
D= { (x,y,2) | x² + y²<1 and z = 0 }
E = { (x,y.2)| z = J + y? – 1 and – 1szs0}
a) Graph C,D, and E so that we can see what's going on. Note that D would be input as just z = 0, cut off by C as its
boundary. You should note that E lines up with C as its boundary also.
b) Find V x F, the curl of F. You will use this below.
Now our goal is to verify the Curl Theorem, and again, we'll do it twice. The Curl Theorem claims that
fF- dr = |v x F) - as = v×F) - as
c) First evaluate the leftmost expression directly, the line integral of F along the closed curve C.
d) Next evaluate the middle expression directly, the flux integral of the curl of F through the disc D. Remember to be
careful with the orientation of the normal vector of D; it must match up with the counterclockwise orientation of C via
the right-hand rule. This should happen "naturally" but double check anyway.
e) Finally evaluate the rightmost expression directly, the flux integral of the curl of F through the cone E. Same warning
about orientation applies here.
Advice: use polar coordinates for both parts d and e. For part e, note that the cone E is just z = r - 1. You should also
use r and e to parameterize that surface. There will be some funky notation issues because r (the parameterization of
the curve C given above) and r = r2 + yz are different. You might see weird things like r, and the problem is fairly
long and difficult no matter what. Keep going – you can do it!
All three answers from c, d, and e should match, verifying that the Curl Theorem is true in this case.
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