Evaluate the flux integral ∫∫s[0,0,yz] dS, where S is the surface with parametric equation x = uv, y = u + v, z = u-v on R: u2+v2 <= 4 and u > 0.
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Evaluate the flux integral ∫∫s[0,0,yz] dS, where S is the surface with parametric equation x = uv, y = u + v, z = u-v on R: u2+v2 <= 4 and u > 0.
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- Determine the area of the parametric surface S with equation Ŕ(u, v) = (u²+v², 2uv, u²—v²), where 0 ≤ ≤ 1 and 0 ≤ u ≤ 1. VLet C be the curve of intersection of the two surfaces x³ + 2xy + yz = 7 and 3x²-yz = 1. Find parametric equations of the tangent line to C at (1,2,1).Find the equation of the tangent plane to the parametric surface x = u², y = u- v, z = v² , at the point (1, 0, 1).
- Consider the line perpendicular to the surface z=x^2+y^2 at the point where x=3 and y=2. Find a vector parametric equation for this line in terms of the parameter t.Integrate F =-(y sin z)i + (x sin z)j + (xy cos z)k around the circle cut from the sphere x2 + y2 + z2 = 5 by the plane z =-1, clockwise as viewed from above.Find a parametrization of the curve of intersection of the surfaces z = x² – y² and z = x2 + xy + 1.
- Show that the line normal to the surface xy + z = 2 at the point (1, 1, 1) passes through the origin.2. Find an equation of the tangent plane to the parametric surface F(u, v) = (, 2uv, uv?) at the point (-2, -4, –4).Find an equation of the tangent plane to the following parametric surface,r(u, v) = (u2 + 6) i + (v3 + 8u) j + (u + 3v) k ,at the point (7, 7, −2).Write the equation in the form ax + by + cz + d = 0, where a, b, c, and d have no common factors. Then enter the values of a, b, c, and d (in that order) into the answer box below, separated with commas.
- Find a set of parametric equations for the tangent line to the curve of intersection of the surfaces z = x2 + y2 , z = 4 − y at the given point (2, −1, 5).Find a set of parametric equations for the tangent line to the curve of intersection of the surfaces x2 + y2 + z2 = 14, x − y − z = 0, at the given point (3, 1, 2).Find a parametric function for the intersection of r – 3y + 4z = 0 and a? + y² = 9.