(1) Let P denote the solid bounded by the surface of the hemisphere z -Vl-r-y? and the cone2y2 and let n denote an outw...
(1) Let P denote the solid bounded by the surface of the hemisphere zV1--y2 and the cone z-Vx2 + y2 and let n denote an outwardly directed unit normal vector. Define the vector field (a) Evaluate the surface integral F nds directly without using Gauss' Divergence T heorem (b) Evaluate thetriplengral IIdiv(F) dV directly without using Gauss Diver- gence Theorem. confirming the result of Gauss' Divergence Theorem for this particular example. (1) Let P denote the solid bounded by the...
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...
Question 3. Let Q be the solid hemisphere bounded by x + y² + 2 = 1 for 2 > 0 and by the plane z = 0, and let F = xi+yi + zk be a vector field. Verify the divergence theorem for Q and F by answering parts (a) and (b) below. Part (a) (5 points). Find the value of the triple integral of the divergence of F over the solid hemisphere Q. Part (b) (10 points). Evaluate...
(7) Let V be the region in R3 enclosed by the surfaces+2 20 and z1. Let S denote the closed surface of V and let n denote the outward unit normal. Calculate the flux of the vector field F(x, y, z) = yi + (r2-zjy + ~2k out of V and verify Gauss Divergence Theorem holds for this case. That is, calculate the flux directly as a surface integral and show it gives the same answer as the triple integral...
I'll ask again, Please DON'T use the divergence theroem, I cant do the surface integral. (7) Let V be the region in R3 enclosed by the surfaces ++22,0 and1. Let S denote the closed surface of V and let n denote the outward unit normal. Calculate the flux of the vector field Fx, y, z)(2 - 2)j 22k out of V and verify Gauss' Divergence Theorem holds for this case. That is, calculate the flux directly as a surface integral...
Let F(x, y, z) (xr,y, z). Compute the outward flux of F: 9y2622 on the bounded region inside of S. However, you may wish to consider the region bounded between S and the sphere of radius 100.) 7/Fthrough the ellipsoid 4c2 36. (Hint: Because F is not continuous at zero, you cannot use the divergence theorem Suppose that E is the unit cube in the first octant and F(z,y, z) = (-x,y, z). Let S be the surface obtained by...
3. F is the vector field The surface S is the boundary of a solid E, where E is bounded by the sphe:93 x2 + y2 + z2 = 4 and x2 + y2 + z2-) for z > 0, Do the following (a) State the defining equation for Gauss' Theorem. (10 points) (b) Show that div F(a+y). (10 points) (c) Use Gauss' Theorem to rewrite the following integral as product of one dimensional integrals. Do not evaluate. (10 points)...
4.8) a) Complete the statement of: The Divergence Theorem: Let D be a closed solid in space bounded by a closed surface s oriented by an outwardly directed unit normal vector n. If F(x, y, z)=(M(x,y,z), N(x, y, z), P(x, y, z)) where M, N, and P have continuous partial derivatives in D, then: D b) Use the Divergence Theorem to write as an iterated integral the flux of F=(x",x’y,x?:) over the closed cylindrical surface whose sides are defined by...
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2 + z2)1/2" a. Evaluate a surface integral to show that SsFonds = 4ta?, where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a)...
2. Follow the steps to verify the Divergence Theorem forF(x, y, z)-(z2, 2y, 49) and the solid cylinder E : r2 + y2 < 4, 0 2. (a) 9 pts] Evaluate F dS directly where S is the closed cylinder S which bounds E oriented outward. Note that S consists of three surfaces: S1 the surface of the cylinder x2 + y-4 for 0 z 2, the disc Di : x2 +92-4 which lies in the plane z 0 and...