Arosinu+rvi-r)for-1 < u < 1 and (r+1 cos ur+1 sin u, Let x(u, e) 9. = (a) Compute the first funda...
16, Let x: U R2-, R, where x(8, φ) (sin θ cos φ, sin θ sin φ, cos θ), be a parametrization of the unit sphere S2. Let and show that a new parametrization of the coordinate neighborhood x(U) = V can be given by y(u, (sech u cos e, sech u sin e, tanh u Prove that in the parametrization y the coefficients of the first fundamental form are Thus, y-1: V : S2 → R2 is a conformal...
5. In class we saw that the function r(u, v) = (sin u, (2 + cos u) cos v, (2 + cos u) sin v), 0<u<27, 050521 parametrizes a torus T, which is depicted below. (a) Calculate ||ru x rull. (b) Show that T is smooth. (c) Find the equation of the tangent plane to T at (0,). (d) Find the surface area of T (e) Earlier in the semester, we observed that a torus can be built out of...
Let F(x, y, z) = sin yi + (x cos y + cos z)j – ysin zk be a vector field in R3. (a) Verify that F is a conservative vector field. (b) Find a potential function f such that F = Vf. (C) Use the fundamental theorem of line integrals to evaluate ScF. dr along the curve C: r(t) = sin ti + tj + 2tk, 0 < t < A/2.
. Compute the curvature and the osculating circle to the curve FC) - (sin(e2), cos(e"), e?) at the point where t In vm. . Compute the curvature and the osculating circle to the curve FC) - (sin(e2), cos(e"), e?) at the point where t In vm.
1L COS v 21) Let H denote the surface parametrized by r(u, )sin, where 7 0S11 land 0 < u < 2T. (a) Compute Tu, Tu, and Tu X T, (b) Compute 1L COS v 21) Let H denote the surface parametrized by r(u, )sin, where 7 0S11 land 0
Question 1. Let y : R -> R' be the parametrised curve 8 (t)= 1+ sin t Cost 5 Cos (a) (2 marks) Show that y is unit speed (7 marks) Find, at each point on the curve, the principal tangent T, principal normal (b) N, binormal B, curvature K, and torsion 7. (c) (3 marks) Show directly that T, N, B satisfy the Frenet-Serret frame equations (d) (3 marks) Show that the image of y lies in a plane...
Let u = u(x,y) and x = x(r,9), y = y(r,). ди ди a. Let x = r cos Q, y = r sin p. Find and a2u ar' 29 ar2 b. u = -x x = r sin 29,y = r tan’ 4, P (1,5). Find ou at the point P. де до
9. Let V(x,y,+) - +w7+ sin(x + y)e'] + cos(x + 2) be a vector field in R'. Compute the curl and divergence of the vector field.
W60. Compute (sin x + cos x)(4 – 2 sin 2x – sin? 2x)e" dx sin 2x where = 6 (0,7) Mihály Be Pirkulyiev Rovsen W38. Let (an)n be a sequence, given by the recurrence: man+1 + (m - 2) an an-1 = 0 where me R is a parameter and the first two terms of (an), are fixed known real numbers. Find me R, so that lim an = 0 n-00 Tad
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T (2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T