4. Consider the surface (cone) S given by (a) Calculate the surface area of S (b) Equipping S wit...
3. Sketch S and compute where S is the part of the cone z-Vx+y* between z-1 andz -3, oriented by the unit normal with negative z-component. S is the oriented surface given by the parametrization ф(II,'')-(11+1, 112-r ,uv) and (11, v) E [0.1] x [0.1] S is upper unit hemisphere, oriented by the unit normal pointing away from the origin. 3. Sketch S and compute where S is the part of the cone z-Vx+y* between z-1 andz -3, oriented by...
(a) Use surface integral(s) to calculate the flux of the vector field or through the given surface. (b) Use the divergence theorem to calculate the flux of the vector field through the given surface. 4. F(x, y, z) =x2yi - 2yzj + x2y2k; S is the surface of the rectangular solid in the first octant bounded by the planes x= 1,y=2, and z=3. Show your work and give an exact answer.
5. Setup (but do not evaluate) one integral (of any type) to find the flux of vector field F through surface S, where S s the unit cube given by 0 < x < 1,0 < y 1.0 < z 1, 5. Setup (but do not evaluate) one integral (of any type) to find the flux of vector field F through surface S, where S s the unit cube given by 0
Il Evaluate the surface integral F.ds for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = y) - zk, S consists of the paraboloid y = x2 + 22,0 Sys1, and the disk x2 + z2 s 1, y = 1. Evaluate the surface integral F.ds for the given vector field F and the oriented surface S....
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...
Show all work and use correct notation for full credit. Stokes' Theorem: Let S be an orientable, piecewise smooth surface, bounded by a simple closed piecewise smooth curve C with positive orientation. Let F be a vector field with component functions that have continuous partial derivatives on an open region in R3 that contains S. Then | | curl(F) . ds F-dr = where curl(F) = ▽ × F. (2 Credits) Let S be the cone given by {(z, y,...
4. Suppose S is the surface z= x² + 4y’, lying beneath the plane z=1. Orient S by taking the inner normal n to pointing in the positive k direction. Calculate the flux integral across S in the direction n for the velocity field of a gas given by V= <y,-X2,xz'>[12]
4. Suppose S is the surface z= x² + 4y? Tying beneath the plane z=1. Orient S by taking the inner normal n to pointing in the positive k direction. Calculate the flux integral across S in the direction n for the velocity field of a gas given by V- <y; -xz,xz> [12]
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
Suppose S is the surface z= x2 + 4y2, lying beneath the plane z=1. Orient S by taking the inner normal n to pointing in the positive k direction. Calculate the flux integral across S in the direction n for the velocity field of a gas given by V= <y, -x,x=2>[12]