4. Let K be the cone with equation z = 4Vr2 + уг, for 0 Compute 4, and let F be the vector field F = <-y,za). z F dS...
(14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the surface of the cone shown at right parameterized by G(r,)-(rcos(0),r sin(0),6-3r) Write the integral F dS using an outward pointing normal in dS terms r and θ. This cone has an open bottom. . The integrand must be fully simplified » Do not evaluate the integral (14 points) Let F be the radial vector field Ft(z, y, z) =zi+w+sk And S be the...
(1) Let G(,y, z) = (x,y, z). Show that there exists no vector field A : R3 -> R3 such that curl(A) Hint: compute its divergence G. (2) Let H R3 -> R3 be given as H(x,y, z) = (1,2,3). Find a vector potential A : R3 -> R3 such that curl(A) smooth function = H. Show that if A is a vector potential for H, then so is A+ Vf, for any f : R5 -> R (3) Let...
Consider the vector field F(x, y, z) = (z arctan(y2), 22 In(22 +1), 32) Let the surface S be the part of the sphere x2 + y2 + x2 = 4 that lies above the plane 2=1 and be oriented downwards. (a) Find the divergence of F. (b) Compute the flux integral SS. F . ñ ds.
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
Computing dS: A student is asked to compute the downward pointing vector surface element ds for the cone z x2 y2 in Cartesian coordinates, parameterized by the vector function: Their work is shown below. There are mistakes in each of the four lines of their work! Correct the mistakes as they occur (Only correct the mistake in the first line it occurs in; ignore it in the following work. This is how I assign partial credit -take off points for...
please help me solve the following question 8. Compute JJ f dS where f(x, y, 2)22+2 and S is the top hemisphere x2 + y2 + Z2, 220. 9. Compute JJ F-n dS where F-: (x, y, z) and s is the cone z2 x2 + y2, 0 S 2 1; with the outward pointing normal. 8. Compute JJ f dS where f(x, y, 2)22+2 and S is the top hemisphere x2 + y2 + Z2, 220. 9. Compute JJ...
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- pointing unit normal. 6-Let F(x, y,z) = yi - xj+zx°y?k. Evaluate (V x F) dS where S is the surfacex2232 = 1, z
3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized by (u,v)-(ucos v, u sin v, hu) x2+y2 a at height h above the xy-plane Z = a V 0<vsa, OSvs 2n, and as is the curve parametrized by ē(f) =(acost,asint, h), 0sis27 as x2+ a 3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized...
Let P denote the solid bounded by the surface of the hemisphere z -vl--g and the cone z-Vr2 + y2 and let n denote an outwardly directed unit normal vector Define the vector field (a) Evaluate the surface inteFn dS directly without using Gauss' Divergence aP Theorem (b) Evaluate the triple integraldiv(F) dV directly without using Gauss' Diver gence Theorem. Note: You should obtain the same answer in (a) and (b) In this question you are confirming the result of...
Let F(x,y,z) = <2y2z, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative, find a function f such that F = Vf and f(1,2,1)= 8. (b) Using the result of part(a), evaluate the line integral of F along the following curve C from (0, 0, 0) to (3.9, 1.4, 2.6). y2 + x4z3 + 2xy(x3 + y4 + 24)1/3 = K ; K is a constant Answer: Next page