10. Consider the surface S parameterized by w r= (cos y, sin v, u + sin...
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...
Question 1 1 pts Let F= (2,0, y) and let S be the oriented surface parameterized by G(u, v) = (u? – v, u, v2) for 0 <u < 12, -1 <u< 4. Calculate | [F. ds. (enter an integer) Question 2 1 pts Calculate (F.ds for the oriented surface F=(y,z,«), plane 6x – 7y+z=1,0 < x <1,0 Sysi, with an upward pointing normal. (enter an integer) Question 3 1 pts Calc F. ds for the oriented surface F =...
3. (3 points) Let the surface S be parametrized by r(u, v) = (bcos u, sin u, v) for (u, v) E D where D = {(u, v) O SUST, SU <3}. Set up the iterated integral, but do not evaluate, the surface area JJsdS (I want the iterated integral for du du, and in that order. Do not even try to evaluate this integral!).
Problem 6. Describe the surface r(u, u)-R cos u x + R sin u ý + uz where 0 < u < 2π and 0 < u-H. and R and H are positive constants. What is the surface element and what is the total surface area? Show that Or/au, or/àv are continuous across the "cut at 2T coS W T
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 =...
(3) Consider f: R3- R3 defined by (u,, w)-f(r, y, :) where u=x w = 3~. Let A = {1 < x < 2, 0 < xy < 2, 0 < z < 1). Write down (i) the derivative Df as a matrix (ii) the Jacobian determinant, (ii) sketch A in (x, y. :)-space, and iv) sketch f(A) in (u. v, w)-space.
in each case: (e) Compute y = sin(z)cos(r) for 0 < z < π/2
8. Solve V?u=0, 2<r<4,0<O<21, (u(2,0) = sin 0, u(4,0) = cos 0,0 5 0 5 21.
Please solve this question The image of the parametrization Ф(u, u)-(a . sin(u) . cos(v), b . sin(u) . sin(v), c . cos(u)) with óくa, 0 < u < π, 0 < v < 2π parametrizes an ellipsoid. a) Show that all the points in the image of Ф satisfy the Cartesian equation of an ellipsoid E 2 b) Show that the image surface is regular at all points c) Write out the integral for its surface area A(E), (Do...
Evaluate the surface integral. y ds, S is the helicoid with vector equation r(u, v) = (u cos(V), u sin(), v), OSUS 4,0 SV S.