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
Find the area of the surface over the given region. Use a computer algebra system to verify your results. The torus r(u, v)-(a + b cos v)cos ui + (a + b cos v)sin uj + b sin vk, where a > b, 0 2 π, b > 0, and 0 2π u v Find the area of the surface over the given region. Use a computer algebra system to verify your results. The torus r(u, v)-(a + b cos...
10. Consider the surface S parameterized by w r= (cos y, sin v, u + sin v), -3 <u <3, 050 < 27 *** (a) Write a linear equation for the tangent plane to the surface at (0,1,1) (b) Compute the surface area of S.
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...
(5) The image of the parametrization Φ(u, u) = (a . sin(u) . cos(u), b . sin(u) . sin(e), c . cos(u)) sin(u sin() cosu with b < a, 0 r, 0 2π parametrizes an ellipsoid. u u a) Show that all the points in the image of Φ satisfy the Cartesian equation of an ellipsoid E b) Show that the image surface is regular at all points. c Write out the integral for its surface area A(E). (Do not...
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...
Time series analysis 2. Set n 100 and generate and plot the time series xt 2 cos(2π.06t) + 3 sin(2π.06t) Ý,-4 cos(2n. 10t) + 5 sin(2m10) z, 6 cos(2π·40t) + 7 sin(2π·40t) (a) Use the periodogram function in R to plot the periodogram of Vi. Can you explain the spikes? (b) Now let wi ~ N(0, 25) be iid and plot the periodogram of the series V +w. Does it still pick out the periodic components? 2. Set n 100...
An ellipsoid, S, is parameterised by r = a sin θ cos φi + a sin θ sin φj + b cos θk 0 ≤ θ ≤ π 0 ≤ φ ≤ 2π i. Find the surface element dS, such that dS points OUT of the ellipsoid. ii. Hence determine the following surface integral over the ellipsoid: //rds JJs
Solve the heat flow problem: ot (x, t) au au (x, t) = 2 (x, t), 0 < x <1, t > 0, a x2 uz(0,t) = uz(1, t) = 0, t> 0, u(a,0) = 1 + 3 cos(TTX) – 2 cos(31x), 0<x< 1.
Problem 4, Find, for 0-x-π, the arc-length of the segment of the curve R(t) = (2 cos t-cos 2t, 2 sin t-sin 2t) corresponding to 0< t < r