2Let S be the positively oriented surface parametrized byR(u, v) = (u, v, u² - v²)where (u, v) are on a quadrilateral region on the uv-plane with vertices at (0, 0),(3,0), (3, 1), and (1, 1). Compute the flux of a fluid with velocity fieldF(x, y, z) = (y, -x, z)across S.

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Answered: 2 Let S be the positively oriented…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let the velocity of a fluid at each point (x,y,z) be given by V(x,y,z) = <x,y,z>. If S is the surface given by z = 1-(x^2 + y^2 ), z is greater or equals to zero Find the flux of V through the surface S in the upward direction
Fs<x, y, z> = < 0, y, -z >, S consists of the paraboloid y = x2 + z2 , 0 < y < 1, and the disk x2 + z2 < 1, y = 1. Evaluate the surface integral 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.
This means we can use the Divergence Theorem to find the flux Let S be the surface of the portion of the solid cylinder of radius 3 centered along the z-axis that lies betweenthe planes z = 0 and z = 4. Find the flux of ~F = x~i + y ~j + z ~k through S, where S is given the outward orientation.
se the Divergence Theorem to evaluate S F · N dS and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y, z) = x3i + x2yj + x2eyk S: z = 4 − y, z = 0, x = 0, x = 3, y = 0
Consider a flow with velocity v→ (x,y,z) and density ρ(x,y,z), such that F→ and let n → =ρv→be given by F→= xi→ +yj→ -2zk→. Let S be the surface x2 + y2 + z2 = 4, z ≥ √2 the normal with n→ component z > 0. Calculate the flux of F→ through S.
Let S be the portion of the cylinder y = ln x in the first octant whose projection parallel to the y-axis onto the xz-plane is the rectangle Rxz: 1 ≤ x ≤ e, 0 ≤ z ≤ 1. Let n be the unit vector normal to S that points away from the xz-plane. Find the flux of F = 2yj + zk through S in the direction of n.
Set up the surface integral to evaluate the flux of the vector field F=<y,x,x+z>, where S is the surface of the cylinder x2+y2=1 and the planes z=-1 and z=2 oriented outward. Please include 3D sketches of the 3 surfaces, sketches of the domain for each surface, normals for each surface. No need to evaluate to integral, just need help setting it up.

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2 Let S be the positively oriented surface parametrized by R(u, v) = (u, v, u² – v²) where (u, v) are on a quadrilateral region on the uv-plane with vertices at (0, 0), (3,0), (3, 1), and (1,1). Compute the flux of a fluid with velocity field F(x, y, z) = (y, -x, z) across S.