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4. Consider the surface of revolution o(u, v) (f(u)cosv, f(u) sin v, g(u)) where uf(u), 0,...
Question 4 (Geodesics on surfaces of revolution) Let S be a surface of revolution and consider...
Question 4 (Geodesics on surfaces of revolution) Let S be a surface of revolution and consider for it the parametrization x(u, v) ((v) cos u, p(v) sin u, ^(v) Assume in addition that (a)2 +()21 (a) Prove that a curve a(t) = x(u(t), v(t)) is a geodesic of S if and only if it satisfies dip 1 ü2 dv p dip p(u)2 0, dv where here and in what follows the dot denotes derivative with respect to t 5 marks...
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ,...
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ, υ), θΕ[0, 27), where f(u) is a positive function. For the following choices of f(u), find the principal, Gaus- sian, and mean curvatures at arbitrary (u, θ), and classify each point on the surface as elliptic, hyperbolic, parabolic, or planar. (a) f(u)u, u E [0, 00) (b) f(uV1 - u2,u e[-1,1].
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field...
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field on a, consider the Darboux frame field T, V, U—where T is the unit tangent of a, U is the surface normal restricted to a, and V = U * T (Fig. 5.34). (a) Show that T' = gV + kU V' =-gT + tU, U' = -KT - tv, 263/518 where k = S(T) · T is the normal curvature k(T) of M...
1. Who's that surface? Consider the function Flu, y) = (v cosu, v sin u, u),...
1. Who's that surface? Consider the function Flu, y) = (v cosu, v sin u, u), 0 Su<27, -2 SU <2. The goal of this problem is to figure out what surface this function parametrizes! (a) Find a parametrization of the coordinate curve with u held constant as u = u. Plot a couple of these curves in 3D to see what they look like. (b) Find a parametrization of the coordinate curve with v held constant as v =...
Find the first quadratic form of a surface called helicoid, r(u, v)-(u cos v, u sin v, av), where...
Find the first quadratic form of a surface called helicoid, r(u, v)-(u cos v, u sin v, av), where a is a constant parameter. Then find the angle of intersection of lines on the surface of helicoid given by equations u v0, u0.
Find the first quadratic form of a surface called helicoid, r(u, v)-(u cos v, u sin v, av), where a is a constant parameter. Then find the angle of intersection of lines on the surface of helicoid...
Let F = <z, 0, y> and let S be the oriented surface parametrized by G(u, v) = (u2 − v, u, v2) for 0 ≤ u ≤ 6, −1 ≤ v ≤ 4. Calculate the normal component of F to the surface at P = (24, 5, 1) = G(...
Let F = <z, 0, y> and let S be the oriented surface parametrized by G(u, v) = (u2 − v, u, v2) for 0 ≤ u ≤ 6, −1 ≤ v ≤ 4. Calculate the normal component of F to the surface at P = (24, 5, 1) = G(5, 1).
3) Consider the vector field F-ra where a is a constant vector and let V be the region in R3 bounded by the surfaces r y24,1, z0. Find the outward flux of F i1n across the closed surface S of V...
3) Consider the vector field F-ra where a is a constant vector and let V be the region in R3 bounded by the surfaces r y24,1, z0. Find the outward flux of F i1n across the closed surface S of V
3) Consider the vector field F-ra where a is a constant vector and let V be the region in R3 bounded by the surfaces r y24,1, z0. Find the outward flux of F i1n across the closed surface S...
3) 7 points - Find the surface area of the surface given parametrically by 7(u, v) = 2 sin u cos vi + 2 sin u sinvj+2cos uk , 0 u π,0 vS2π 3) 7 points - Find the surface area of the surface g...
3) 7 points - Find the surface area of the surface given parametrically by 7(u, v) = 2 sin u cos vi + 2 sin u sinvj+2cos uk , 0 u π,0 vS2π
3) 7 points - Find the surface area of the surface given parametrically by 7(u, v) = 2 sin u cos vi + 2 sin u sinvj+2cos uk , 0 u π,0 vS2π
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [...
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
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...
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...
Question 4 (Geodesics on surfaces of revolution) Let S be a surface of revolution and consider for it the parametrization x(u, v) ((v) cos u, p(v) sin u, ^(v) Assume in addition that (a)2 +()21 (a) Prove that a curve a(t) = x(u(t), v(t)) is a geodesic of S if and only if it satisfies dip 1 ü2 dv p dip p(u)2 0, dv where here and in what follows the dot denotes derivative with respect to t 5 marks...
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ, υ), θΕ[0, 27), where f(u) is a positive function. For the following choices of f(u), find the principal, Gaus- sian, and mean curvatures at arbitrary (u, θ), and classify each point on the surface as elliptic, hyperbolic, parabolic, or planar. (a) f(u)u, u E [0, 00) (b) f(uV1 - u2,u e[-1,1].
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field on a, consider the Darboux frame field T, V, U—where T is the unit tangent of a, U is the surface normal restricted to a, and V = U * T (Fig. 5.34). (a) Show that T' = gV + kU V' =-gT + tU, U' = -KT - tv, 263/518 where k = S(T) · T is the normal curvature k(T) of M...
1. Who's that surface? Consider the function Flu, y) = (v cosu, v sin u, u), 0 Su<27, -2 SU <2. The goal of this problem is to figure out what surface this function parametrizes! (a) Find a parametrization of the coordinate curve with u held constant as u = u. Plot a couple of these curves in 3D to see what they look like. (b) Find a parametrization of the coordinate curve with v held constant as v =...
Find the first quadratic form of a surface called helicoid, r(u, v)-(u cos v, u sin v, av), where a is a constant parameter. Then find the angle of intersection of lines on the surface of helicoid given by equations u v0, u0.
Find the first quadratic form of a surface called helicoid, r(u, v)-(u cos v, u sin v, av), where a is a constant parameter. Then find the angle of intersection of lines on the surface of helicoid...
3) Consider the vector field F-ra where a is a constant vector and let V be the region in R3 bounded by the surfaces r y24,1, z0. Find the outward flux of F i1n across the closed surface S of V
3) Consider the vector field F-ra where a is a constant vector and let V be the region in R3 bounded by the surfaces r y24,1, z0. Find the outward flux of F i1n across the closed surface S...
3) 7 points - Find the surface area of the surface given parametrically by 7(u, v) = 2 sin u cos vi + 2 sin u sinvj+2cos uk , 0 u π,0 vS2π
3) 7 points - Find the surface area of the surface given parametrically by 7(u, v) = 2 sin u cos vi + 2 sin u sinvj+2cos uk , 0 u π,0 vS2π
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
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...
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