Compute the differential of surface area for the surface S described by the given parametrization...
Sketch S and compute integral of ω where S is the oriented surface given by the parametrization Ф(u, v) (11+1, 112-r ,in) and (u, v) [0.1]х [0,1] S is the oriented surface given by the parametrization Ф(u, v) (11+1, 112-r ,in) and (u, v) [0.1]х [0,1]
Calculus . Let h(x, y) be a smooth parametrization on a region H for a surface S in R3. Suppose there is a continuous transform F :R + H, (u, v) + x(u, v), y(u, v)) such that F is one-to-one on the interior of the region R and r;=ho F is a smooth parametrization on R for S. Show that 9 S/ \ru xroldA= S/ \he x hy|dA= A(s where A(S) is the area of S. (15 pts] 9
Let S be the ‘football’ surface formed by rotating the curve y = 0, x = cos z for z ∈ [−π/2, π/2], around the z-axis. Find a parametrization for S, and compute its surface area. Please answer in full With full instructions. Let S be the 'football, surface formed by rotating the curve y = 0, x-cosz for-E-π/2, π/2], around the z-axis. Find a parametrization for S, and compute its surface area 3 Let S be the 'football, surface...
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...
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.
3) 7 points - Find the surface area of the surface given parametrically by 7(u, v) = 2 sin u cos vi + 2 sin u sinvj+2cos uk , 0 u π,0 vS2π 3) 7 points - Find the surface area of the surface given parametrically by 7(u, v) = 2 sin u cos vi + 2 sin u sinvj+2cos uk , 0 u π,0 vS2π
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...
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.
Consider the mountain known as Mount Wolf, whose surface can be described by the parametrization (u, v) = (u, v, 7565 - 0.0202 - 0.0312) with v2 + v2 = 10,000, where distance is measured in meters. The air pressure P(x, y, z) in the neighborhood of Mount Wolf is given by P(x, y, z) = 32el-7x2 + 4y2 + 2z). Then the composition Q(u, v) = (Por)(u, v) gives the pressure on the surface of the mountain in terms...
#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...