The tangent plane at a point Po(f(uo.VO) 9 (uo.vo) h(uo,VO)) on a parametrized surface r(u,v) =...
(2) Let S be the surface parametrized by r(u, v) = (u? – 12)i + (u + v)j + (u? + 3v)k. (a) Find a normal vector to S at the point (3,1,1). (b) Find an equation of the tangent plane to S at (3, 1, 1).
4. (1 pt) Calculate Tu, T, and n(u, v) for the parametrized surface at the given point Then find the equation of the tangent plane to the surface at that point Ф(и, у) %3D (2и + v, и — 4v, 5и); Ти The tangent plane: V u=4, v6 , n(u,v) TV =9z
7. Find an equation of the tangent plane to the given parametric surface r(u, v) = uvi+u sin(n)j + v cos(u)k, at u = 0, v = . 8. Find the area of the part of the surface 2 = 2 + 5x + 2y that lies above the triangle with vertices (0.0), (0,1), and (2,1).
Find an equation of the tangent plane to the following parametric surface, r(u, v) = (u2 + 9) i + (v3 + 6u) j + (u + 2v) k , at the point (10, 5, −1). 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...
Help would be greatly appreciated!! 1. Let S be the surface in R3 parametrized by the vector function ru, v)(,-v, v+ 2u) with domain D-{(u, u) : 0 u 1,0 u 2). This surface is a plane segment shaped like a parallelogram, and its boundary aS (with positive orientation) is made up of four line segments. Compute the line integral fos F -dr where F(z, y, z) = 〈エ2018 + y, 2r, r2-Ins). Hint: use Stokes' theorem to transform this...
Assume that is the parametric surface r= x(u, v) i + y(u, v) j + z(u, v) k where (u, v) varies over a region R. Express the surface integral 116.3.2) as as a double integral with variables of integration u and v. a (x, y) a(u, v) du dy ru Хry dy du l|ru Xr, || f (x (u, v),y(u, v),z (u, v)) 1(xu, Wsx,y,z) Mos u.v.gou,» @ +()*+1 li ser(u, v),y(u, v),z (u, v) Date f (u, v,...
Calculate Tr, T. and N(r, θ) for the parametrized surface at the given point. | I θ . r ., G(r, θ)-(r cos(9), r sin(θ), 1-r2); 16' 4 6' 4 6' 4 Find the equation of the tangent plane to the surface at that point. Calculate Tr, T. and N(r, θ) for the parametrized surface at the given point. | I θ . r ., G(r, θ)-(r cos(9), r sin(θ), 1-r2); 16' 4 6' 4 6' 4 Find the equation...
4. (4 pts) Consider the surface z=x2y+y3.(a) Find the normal direction of the tangent plane to the surface through (1,1,2).(b) Find the equation of the tangent plane in (a).(c) Determine the value a so that the vector−→v=−−→i+ 2−→j+a−→k is parallel to the tangent plane in (a).(d) Find the equation of the tangent line to the level curve of the surface through (1,1). 4. (4 pts) Consider the surface z = z2y + y). (a) Find the normal direction of the...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0 Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
Find an equation of the tangent plane to the given parametric surface at the specified point.r(u, v) = u cos vi + u sin vj + vk; u = 9, v = p/3