1L COS v 21) Let H denote the surface parametrized by r(u, )sin, where 7 0S11 land 0 < u < 2T. (a...
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
Let M be the surface parametrized by T: (1, 0) R → R3 (u, v) = (ucov, usin 0,0 + 8"}2+1]" d) 1 Compute the mean curvature of M.
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...
If u(t) = sin(2t), cos(2t), t and v(t) = t, cos(2t), sin(2t) , use Formula 5 of this theorem to find d dt u(t) × v(t) .
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...
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...
(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).
#17 and #21 17) r= ( 2 cosht cos 0,3 cosht sin o, sinht) (hyperboloid) 18. r= ( 2 cosht cos , sinht, 3 cosht sin o ) (hyperboloid) a ) (hyperbolic parboloid) x² y ² 19. r= ( x,y, 4 y2 22 20. r= ( , y, > (hyperbolic parboloid) 25 16 21. r= ( 2u cosh v, 3u sinh v, u? ) (hyperbolic parboloid) Surface Area In Exercises 23-42, compute the surface area of the surface S parametrized...
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.