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Stokes's Theorem / The Curl Theorem 2. Consider the vector field F(x,y,2) = (yz²,x,x +y) and the closed curve C: r(t) = (cos t,sin t,0) 0sts 2n 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< 1and z = 0} E = { (x,y.2)|2= J +y?-1 and – 13z<0} 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 Vx 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 -dr = (V × F) « (V x F) - ds c) First evaluate the leftmost expression directly, the line integral of F along the closed curve C.

Stokes's Theorem / The Curl Theorem 2. Consider the vector field F(x,y,2) = (yz²,x,x +y) and the closed curve C: r(t) = (cos t,sin t,0) 0sts 2n 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< 1and z = 0} E = { (x,y.2)|2= J +y?-1 and – 13z<0} 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 Vx 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 -dr = (V × F) « (V x F) - ds c) First evaluate the leftmost expression directly, the line integral of F along the closed curve C.

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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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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