10. Let S be a regular surface with E = G = (1 + u2 + U2)2, F = 0 and e = 2=-g,f=0. (a) Find the ...
Let F = <z, 0, y> and let S be the oriented surface parametrized by G(u, v) = (u2 − v, u, v2) for 0 ≤ u ≤ 6, −1 ≤ v ≤ 4. Calculate the normal component of F to the surface at P = (24, 5, 1) = G(5, 1).
surface patch for S. regular surface and f: S Ra smooth EXERCISE 3.44. Let S be a function. Assume that the point p e S is a critical point of f, which means that dfp(v) 0 for all v e TpS. Define the Hessian of f atp in the direction v as Hess(f)p(v) (foy)"(0), where y is a regular curve in S with y(0) = p and y'(0) = v. Prove that the Hessian is well defined in the sense...
3. A general surface of revolution is r(u, θ)-(f( u') cos θ , f(u) sin θ, υ), θΕ[0, 27), where f(u) is a positive function. For the following choices of f(u), find the principal, Gaus- sian, and mean curvatures at arbitrary (u, θ), and classify each point on the surface as elliptic, hyperbolic, parabolic, or planar. (a) f(u)u, u E [0, 00) (b) f(uV1 - u2,u e[-1,1].
2. Find the principal curvatures and principal directions of the surface obtained by rotating the curve x coshz in the z-plane around the z-axis 2. Find the principal curvatures and principal directions of the surface obtained by rotating the curve x coshz in the z-plane around the z-axis
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
4. Consider the surface of revolution o(u, v) (f(u)cosv, f(u) sin v, g(u)) where uf(u), 0, g(u)) is the unit-speed regular curve in R3, Find the normal curvature of meridian v constant and geodesic curvatures of a parallel u=constant. 4. Consider the surface of revolution o(u, v) (f(u)cosv, f(u) sin v, g(u)) where uf(u), 0, g(u)) is the unit-speed regular curve in R3, Find the normal curvature of meridian v constant and geodesic curvatures of a parallel u=constant.
Let F : R3 → R3 be defined by F(p) = cp where c 〉 0 is a constant. Let Si C R3 be a regular, orientable surface and let S2 F(S). Show that S2 is a regular, orientable surface and write Gauss and mean curvature K2, H2 of S2 interms of Gauss and mean curvature K1, Hi of S. Let F : R3 → R3 be defined by F(p) = cp where c 〉 0 is a constant. Let...
7. Find the surface area of the surface r(u, u) = u ui + (u + u)j + (u-u) k, u2 +02-1 V/16-x2-y2 with upward orientation and let 8. Let S be the hemisphere 2 F(x, y,z)-yitj+3z k. Calculate JJs F dS, the flux of F across S 7. Find the surface area of the surface r(u, u) = u ui + (u + u)j + (u-u) k, u2 +02-1 V/16-x2-y2 with upward orientation and let 8. Let S be...
Exercise 3. Let u2= (5) C) V2 = V1 = and E u1, u2},F = {v1,v2} be two ordered bases for R2. Let also 5 (i) Find the coordinate vectors of [x]E and [x\f. (ii) Find the transition matrix S from the basis E to F. (ii) Verify that [x]f = S[r]E Exercise 3. Let u2= (5) C) V2 = V1 = and E u1, u2},F = {v1,v2} be two ordered bases for R2. Let also 5 (i) Find the...
Exercise 4.5.3. Let G-(g g 1 be a group of order 2 and V a CG-module of Let u +202 +2,u2 2v1 - 2 +2vs,u vector space spanned by ui, for i-1,2,3 2v - 202 +vs, and hence U the (i) Prove that U is a CG-submodule of V fori 1,2,3, and that (ii) Let λ C and u-ul + U2 + λν3 V. Find the value(s) of λ for which the subspace U spanned by u is a CG-submodule...