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
pls show the work clearly 9. Find | V x F ñds where F =< 22,4x, 3y >, the surface S is the cap of the sphere S x2 + y2 + z2 = 169 above xy-plane and the boundary curve C is the boundary of S.
PROBLEM 2 Consider the family of circles P = {C, TER>0}, where Cr = {(x, y) R2 | x2 + y2 = p2} is a circle of radius r > 0. Prove that P is a partition of R2. State an equivalence relation induced by this partition. Hint: What is a property that is True for all points in a fixed circle?
7. (5 marks) Consider a smooth function u(x, y) satisfying: Urx + Uyy + Uzy > 0 in 12. . Show that u atains its maximum on the month ago Show that u attains its maximum on the boundary an.
What is the solution of day 2 dy 1(1+1) dx² + xăx x² y = f(x = a) (a > 0). on the interval 0<x< 0, subject to the boundary conditions y(0) = y(0) = 0? / is a positive integer.
6.62. Let Yi < Y2 < . . . < Y, be the order statistics of a random sample of size n from the distribution having p.df.f(x)-2x/g, 0<x <θ, zero elsewhere (a) If 0 < c < 1, show that Pr (c < Y,/θ < 1)-1-eM (b) If n=5 and if the observed value of Y, is 1.8, find a 99 percent confidence interval for 0.
Could someone explain how these to get these phase portraits by hand with ẋ=y and ẏ=ax-x^2 especially for a=0 case where you have eigenvalues all equal to zero? 6.5.4 a>0 Sketch the phase portrait for the system x = ax-x, for a < 0, a = 0, and For a -(0 We were unable to transcribe this imageFor a>0 ES CS
For f(x, y) = k(x2 + y2), 0<x< 1 and 0 <y<1 and 0 elsewhere: a) Find k. b) Are X and Y independent? c) Find P(X<0.5, Y>0.5), P( X = 0.5, Y>0.5).
Let X and Y b Var(Y) (1) If a, b,c and d are fixed real numbers, = E(X), μγ E (Y),咳= Var(X) and e ranclom variables. with y a) show Cov(aX +b, cY +d)- ac Cov(X,Y) (b) show Corr(aX + b, cY + d)-PXY for a > 0 and c > 0.
Please help me solve this differential Equation show all steps Find a continuous solution satisfying +y-f(x), where f() Ji 10 { 0<r<1 > 1 and y(0) -0.