6-Let F(x, y,z) = yi - xj+zx°y?k. Evaluate (V x F) dS where S is the surfacex2232 = 1, z < 0 oriented by the upward-...
F-dS where S is the cylinder x? +-2, 0 s y < 2 oriented by the unit normal 5- Let F(x,y,z)= (-6x,0,-62). Evaluate pointing out of the cylinder. 6-Let F(x, y,2)- yi- xj +zx°y?k. Evaluate (Vx F) . dS where S is the surface x2+y+32 - 1, z <0 oriented by the upward- pointing unit normal. F-dS where S is the cylinder x? +-2, 0 s y
5- Let F(x, y, z)= (-6x,0,-6z). Evaluate F.dS where S is the cylinder x2z2 = 2, 0 < y < 2 oriented by the unit normal pointing out of the cylinder. 5- Let F(x, y, z)= (-6x,0,-6z). Evaluate F.dS where S is the cylinder x2z2 = 2, 0
5. Calculate the surface area of the portion of the sphere x2+y2+12-4 between the planes z-1 and z ะไ 6. Evaluate (xyz) dS, where S is the portion of the plane 2x+2y+z-2 that lies in the first octant. 7. Evaluate F. ds. a) F = yli + xzj-k through the cone z = VF+ア0s z 4 with normal pointing away from the z-axis. b) F-yi+xj+ek where S is the portion of the cylinder+y9, 0szs3, 0s r and O s y...
Question 1 1 pts Let F= (2,0, y) and let S be the oriented surface parameterized by G(u, v) = (u? – v, u, v2) for 0 <u < 12, -1 <u< 4. Calculate | [F. ds. (enter an integer) Question 2 1 pts Calculate (F.ds for the oriented surface F=(y,z,«), plane 6x – 7y+z=1,0 < x <1,0 Sysi, with an upward pointing normal. (enter an integer) Question 3 1 pts Calc F. ds for the oriented surface F =...
(2) Let F zi + xj+yk and consider the integral vx Fi n dS where S is the surface of the paraboloid z = 1-x2-y2 corresponding to 0, and n is a unit normal vector to S in the positive z-direction. (a) Apply Stokes' theorem to evaluate the integral. b) Evaluate the integral directly over the surface S. (c) Evaluate the integral directly over the new surface S which is given by the disk (2) Let F zi + xj+yk...
2. Evaluate 1,(1,0, 2) . ds, where s is the cone z = VE4y2 with 0 < z < 2, Upward 1,0,2) ds, where S is the pointing normal. 3. Use a surface integral to find the area of the region of the plane z2y +3 with 2. Evaluate 1,(1,0, 2) . ds, where s is the cone z = VE4y2 with 0
1. Let F(x, y, z) = (-y + ,2-2,2-y), and let S be the surface of the paraboloid 2 = 9-32 - v2 for 2 > 0. oriented by an upward pointing normal vector. Note that the boundary of S is C, the circle of radius 3 in the xy-plane. Verify Stokes' Theorem by computing both sides of the equality: (a) (1 Credit) || (D x F). ds (b) (1 Credit) $F. dr
(6) Show that F(x, y) = (x+y)i + (**)is conservative. (a) Then find such that S = F (potential function). (5) Use the results in part(a) to cakulae ( F. ds along C which the curve y = a* from (0,0) to (2,16). (2) Use Green's Theorem to evaluate 1. F. ds. F(1,y) =(yº+sin(26))i + (2xy2 + cos y)and C is the unit circle oriented counter clockwise (6) Evaluate the surface integral || 9. ds. F(x,y,z) = xi +++where S...
Find the upward flux of F=−yi+xj+12kF=−yi+xj+12k across the part of the spherical surface GG determined by z=f(x,y)=12−x2−y2‾‾‾‾‾‾‾‾‾‾‾‾√,0≤x2+y2≤5 Find the upward flux of F =-yi + xj + 12k across the part of the spherical surface G determined by 30 pi Find the upward flux of F =-yi + xj + 12k across the part of the spherical surface G determined by 30 pi
e.g.4 Evaluate JJs F dS, where j + sin(zy)k and S is the surface of the region E bounded by the parabolic cylinder z- 1 a2 and the planes z-0,y-0, and y + z-2. e.g.4 Evaluate JJs F dS, where j + sin(zy)k and S is the surface of the region E bounded by the parabolic cylinder z- 1 a2 and the planes z-0,y-0, and y + z-2.