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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
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)

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.
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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