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.
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Vectors In Two And Three Dimensions
Section9.FOM: Focus On Modeling: Vectors Fields
Problem 11P
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