Let S be the negatively oriented surface defined by R(u, v) = (u², -uv, v²), where (u, v) € [0.2] × [0, 1]. Set up a double integral equal to the flux of F(x, y, z) = (y. -rz, y) across S.

Q: Compute the flux of F¹ = 5(x + z)i + 5j + 5zk through the surface S given by y = x² + z², with 0 ≤ y…

A: The given surface is  and .The flux is given by  where  is the vector function that describes the…

Q: Compute the flux of F = xi+yj+zk through the curved surface of the cylinder x² + y² = 1 bounded…

A: The given vector field is F→=xi+yj+zk. The goal is to evaluate the flux or surface integral of the…

Q: Find the flux of F(x, y, z) = yi - xj + 2k across S, where S is the helicoid with vector equation…

A:

Q: Find the surface area of the helicoid (spiral ramp) with vector equation F(u, v) = (u cos v, u sin…

A: The surface area of the helicoid can be computed using the surface integral formula:   ∫∫Rru×rvdA…

Q: 2). Find the flux of F(x, y, z) = (0, yz, z) outward through the surface S cut from the cylinder y +…

A:

Q: Calculate ff f(x, y, z) dS For Part of the surface x = 2³, where 0 ≤ a, y ≤ 13-1; f(x, y, z) = I s…

A:

Q: Evaluate the line integral faF · dr, where F(x, y, z) = -2xi – 5yj + 5zk and C is given by the…

A:

Q: Find the flux of G(x, y, z) = (xz)i+ (yz)j+(z²)k outward across the part of the sphere x² + y² +z² =…

A: Here, surface, S={(x,y,z): x2+y2+z2=9, and 1≤z≤2}. For z=1, we get x2+y2=8 and for z=2, we get…

Q: Let S be the surface given by x² + y²+2z² = 1, x > 0, y ≥ 0, z ≥ 0. Find the flux of (z, 0, 0)…

A: Step 1:The surface S given by x2+y2+2z2=1 is an ellipsoid, and we are considering the upper half…

Q: Find the flux of F across S, 1.1. S F. N dS where N is the upward unit normal vector to S. F(x, y,…

A:

Q: Verify Stokes' Theorum for F(x,y,z) = (y,-x,yz) if S is the paraboloid z = x2 + y2 with x2 + y2 = 1,…

A: Given  Fx,y,z=y,-x,yz Then, F→=yi^-xj^+yzk^