Prove X, Y, Z, JER. XKY Prove Z <# X AND Y<j, if and only if (x,y) [z,j]
1. Find (2.rº + y) DV where Q = { (z,y,z) | 0<<<3, -2<y<1, 1<2<2} 1 / 12
5. If a, b E R, prove that abl < (a + b^).
2. Suppose X and Y have the joint pdf fxy(x, y) = e-(x+y), 0 < x < 00, 0 < y < 0o, zero elsewhere. (a) Find the pdf of Z = X+Y. (b) Find the moment generating function of Z.
3. Consider the vector field F(x,y) = (27x D = {(1,y): 0 < rº + y2 <2}. +ya) defined on the region D where a) Directly compute SF. Tds using the definition of the line integral, where C is the unit circle oriented counterclockwise. b). Use Theorem 3.3 (Test for Conservative Vector Fields) from the text to determine if F is conservative. Is your answer consistent with part a)? If not, what is the source of the discrepancy?
1. Let a=(ay, ay) and b= (6,62) be vectors in Rº. a. Verify that <a, b >= 20,62 +5 a,b2 satisfies the inner product axioms. b. a=(-1,3), b= (2,5) Find da,b).
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
[3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that \f'(x) < 1/(1 - 1z| for all z e D[0, 1]. [3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that f'(x) < 1/(1-1-12 for all z e D[0, 1]
Evaluate SSJ (+y– 32) 2V where E = {(x, y, z)| - 55y<0,0 < x <y, 0 <z<x+y?}
[3] 5. Suppose that f: D[0,1] for all z E D[0, 1] D[0,1] is holomorphic, prove that \f'(z) < 1/(1 - 121)2