Vector Fields problems. Question 6 Let f (x, y) = x3 + 4xy + 2y2 and...
7. The vector field F =< 3x2z In y + ze+2 +20, - 3y?, x° In y + ce2 +423 > is conservative. Find a potential function f(x, y, z) such that F=Vf. Y
8), Let X and Y be continuous random variables with joint density function f(x,y)-4xy for 0 < x < y < 1 Otherwise What is the joint density of U and V Y
cense. This license can only be 5. Given f (x,y) = 12 + x2 + 2y2 the point P(2,1)and vector u =< a. Find the directional derivative at the point in the direction of u b. Find the Gradiant
Let F(x,y,z) = <2y2z, 4xyz, 2xy2> be a vector field. (a) Knowing that F is conservative, find a function f such that F = Vf and f(1,2,1)= 8. (b) Using the result of part(a), evaluate the line integral of F along the following curve C from (0, 0, 0) to (3.9, 1.4, 2.6). y2 + x4z3 + 2xy(x3 + y4 + 24)1/3 = K ; K is a constant Answer: Next page
(7 pts.) Let f(x, y, z) = "y and let R be the region {(x, y, z) |2 < x < 4,0 Sy < 3,15 zse}. 2 Evaluate | $180,0,.2) av. R
Question 10 Compute the flux of the vector fields F(x, y, z) =< x, y2,1 > across the portion of the plane r+y+z=1 on the first octant, with orientation pointing toward the positive x direction. (Do not use Stokes' theorem)
24. Let X and Y be continuous random variables with joint density function 4xy for 0 < x, y 1 f(x, y) otherwise. What is the probability of the event X given that Y ?
Calculate the work done by the vector field F(x,y)=4xy, 2x2 along a smooth, simple curve from point (3, −1) to point (4, 2) We were unable to transcribe this imageWe were unable to transcribe this image
Let X and Y have join density 6 f(x, y) =-(x + y)2, 0 < x < 1, 0 < y < 1
QUESTION 12 Let the random variable X and Y have the joint p.d.f. f(x,y) =(zy for 0< <2, 0 < y <2, and z<y otherwise Find P(0KY <1) 16 QUESTION 13 R eter to question 12. Find P(o < x <3I Y-1).