Obtain a parametrization for the surface z = Vx2 + y2, z < 5. Answer 10...
(7.5 points) Let C be the oriented closed space curve traced out by the parametrization r(t) = (cost, sint, sin 2t), 0<t<27 and let v be the vector field in space defined by v(x, y, z) = (et - yº, ey + r), e) (a) Show that C lies on the cylinder x2 + y2 = 1 and the surface z = 2cy. (b) This implies that C can be seen as the boundary of the surface S which is...
Find a parametrization of the surface. The portion of the sphere x² + y2 + z = 2 between the planes z = and z= - 2 2 What is the correct parameterization? Select the correct choice below and fill in the answer boxes within your choice. (Type exact answers.) O A. r(0,0) = i+ k, sos sos sps sos OB. r(0,0) = O C. rq,0) = sos sos OD. r(0,0) =i+j+ k sos 1.ses
Let S be the surface of the 'liptic paraboloid z = 4 - 22 - y2 above the plane z = 0, and with upward orientation. Let F =< yetan(z), -xcos > be a vector field in R3. 9 + Use Stoke's Theorem to compute: Sf curlĒ. ds. S
2. Let I be the surface of the cone z = V x2 + y2 (without the top) between planes z = 0 and z = 2. Let F =< x,y,z2 >. Calculate the upward directed flux SS FdS (a) Using the Divergence Theorem. (10 points) (b) Without using the Divergence Theorem. (20 points)
The solid S sits below the plane z = 2x + 5 and above the region in the xy-plane where 1 < x2 + y2 = 4 and x + y < 0. The volume of S is:
F(x,y,z) =< P, Q, R >=< xz, yz, 2z2 > S: Bounded by z = 1 – x2 - y2 and z = 0) Flux =SS F ñds S (8a) Find the Flux of the vector field F through this closed surface.
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the boundary of S with counterclockwise orientation when looking down from the z-axis. Verify Stokes' Theorem as follows. (a) (i) Sketch the surface S and the curve C. Indicate the orientation of C (ii) Use the...
4. Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t<2n. (a) [5 pts] Show that this curve C lies on the surface S defined by z = 2.cy. F. dr (b) (20 pts] By using Stokes' Theorem, evaluate the line integral| " where F(t,y,z) = (y2 + cos z)i + (sin y+z)j + tk
Begin with the paraboloid z = 22 + y2 for 0 < < 4, and slice it with the plane y = 0. Let S be the surface that remains for y> 0 (excluding the planar surface in xz-plane) oriented downward (i.e. n3 <0). Let C be the Semicircle and the line segment in the plane z = 4 with counterclockwise orientation and F =< 2z+y, 2x + z, 2y + 2 > ZA C 4 S 2 = x2...
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 =...