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...
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)...
please i need the question 15 for the detailed proof and explaination ! thanks ! 233 42 Isometries, Conformal Maps 14, we say that a differentiable map ф: S,--S2 preserves angles when for every p e Si and every pair vi, v2 E T (S,) we have cos(u, 2) cos(dp, (vi). do,()). Prove that pis locally conformal if and only if it preserves angles. 15. Letp: R2 R2 be given by ф(x, y)-(u (x, y), u(x, y), where u and...
V X2 + y2 and θ u(r(z, y), θ(x, y))--sech2 r tanh r sin θ 6. [Sec. I 1.5] Letr tan l (y/z) be the usual polar rectangular coordinates relationships. Furthermore, define and u(r(z, y),θ(z, y)) sech2 r tanh r cos θ Show that tanh r
(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
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a 5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a
3.58 (a) If U(x, y, z) = xy72, find ▽U and V2U. (b) If V(p, φ, z)- P (c) If W"(r, θ, φ.)-z? sin θ cos φ, find W and VzW. sın, find wandV2V.
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...
(a) Let θ : R-+ R be a smooth function. Find the (signed) curvature of the curve a:R- R2 given by cos(θ(t)) dt,I α(s) sin(θ(t)) dt Use your result to give another geometric interpretation to the (signed) curva- ture and its sign? to) rindy,R-- parmetrised with unit speed suchhat y -0and kt) - s for all seR. (a) Let θ : R-+ R be a smooth function. Find the (signed) curvature of the curve a:R- R2 given by cos(θ(t)) dt,I...
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
arosinu+rvi-r)for-1 < u < 1 and (r+1 cos ur+1 sin u, Let x(u, e) 9. = (a) Compute the first fundamental form of S (b) Compute the Christoffel symbols of S (c) Compute the Gaussian curvature of S (d) For which to is the curve a(t) = x(t,%) a godesíc. arosinu+rvi-r)for-1