(6) Show that the semicircle C = {(x,y) = R2 | + y2 = 1, y > 0} is a 1-dimensional manifold with boundary and the hemisphere D= {(x, y, z) | 22 + y2 + z2 = 1, 2 > 0} is a 2-dimensional manifold with boundary. (7) Suppose X is an n-dimensional manifold with boundary. Let ax denote the set of points in the boundary of X. Show that ax is an (n-1)-dimensional manifold.
Which statement is false? o 3 x E R and SY ER such that (a +y=1) 1 (x - y = 5). O x E R and Sye R such that (x + y = 1) ^ (3x + 3y = 4). O V x ER, SY ER such that x+y=1. O 3 x E Z and 3 y E Z such that (x+y=1) 1 (x – y = 5).
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that: 1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...
(e) Let x = (T1,T2, . . . ,xn),y=(y1,y2, . . . ,Un) ER" (i) Show that for any λ E R: 3 where llxll = 1/(x, x). x, y (ii) Use (7) for λ =- to show: 1a1 with equality, if and only if, there exists a λ E R such that y = 1x.
1. Consider the unit circle: (x,y) : x2 y2 = 1. T. Let f R2 ->R be defined by f(x,y) = x2-y, and let F : R2 -> R be defined by F(x, y) Compute the integral of f and F around the unit circle. For the integral of F, proceed in the standard (anticlockwise) direction
4. Let = 0 , 4r + 2y+-2). M={(x,y,z) € R' | - Show that A/ is a one dimensional manifold and find the maximum and minimum values of SIM where f(x,y, z) = ry + z. 4. Let = 0 , 4r + 2y+-2). M={(x,y,z) € R' | - Show that A/ is a one dimensional manifold and find the maximum and minimum values of SIM where f(x,y, z) = ry + z.
4. A random point (X, Y ) is chosen uniformly from within the unit disk in R2, {(x, y)|x2+y2< 1} (a) Let (R, O) denote the polar coordinates of the point (X,Y). Find the joint p.d.f. of R and . Compute the covariance between R and 0. Are R and e are independent? (b) Find E(XI{Y > 0}) and E(Y|{Y > 0}) (c) Compute the covariance between X and Y, Cov(X,Y). Are X and Y are independent? 4. A random...
1. Find the absolute maximum and minimum values of f(r,y) = x2+y2+5y on the disc {(x, y) | x2+y2 < 4}, and identify the points where these values are attained 2. Find the absolute maximum and minimum values of f(x, y) = x3 - 3x - y* + 12y on the closed region bounded by the quadrilateral with vertices at (0,0), (2,2), (2,3), (0,3), and identify the points where these values are attained. 3. A rectangular box is to have...
Problem 1: Consider the following problem x+y+1=1 x2 +y2+z2 =1 max f(x ,y,z)=er+y+1 subject to (a) Solve the problem. (b) Replace the constraints byx+y+1=1.02 and x2+y2+Z2-0.98. What is the approximate change in the optimal value of the objective function? (c) Classify the candidate points for optimality in the local optimization problem.
2. Let R be the region R = {(X,Y)|X2 + y2 < 2} and let (X,Y) be a pair of random variables that is distributed uniformly on this region. That is fx,y(x, y) is constant in this region and 0 elsewhere. State the sample space and find the probability that the random variable x2 + y2 is less than 1, P[X2 +Y? < 1].