MATH 227 Fall 2008Test 4 1.(20pts)(a) Show that F(x,y,z) = (y2+2x)i + (2xy+z2+2y)j +(2yz+2z)k is a conservative vector field in R3 by computing curl(F). curl(F) = det ? ? i j k ?x ?y ?z y2 + 2x 2xy + z2 + 2y 2yz + 2z ? ? = < 2z ?2z,?(0?0),2y ?2y > =< 0,0,0 > (b) Find a potential function, f , for F. fx(x,y,z) = y2 + 2x , so f(x,y,z) = xy2 + x2 + C(y,z). Now ? ?y(xy 2 + x2 + C(y,z)) = 2xy + z2 + 2y =? 2xy + ? ?yC(y,z) = 2xy + z 2 + 2y. Thus C(y,z) = yz2 + y2 + D(z). So ??z(xy2 + x2 + yz2 + y2 + D(z)) = 2yz + 2z =? ddzD(z) = 2z and D(z) = z2 + d, where d is a constant. We conclude that ?f(x,y,z) = F(x,y,z), where f(x,y,z) = xy2 + x2 + yz2 + y2 + z2 + d (c) Find the work done by F on a particle which moves on any curve joining (0,0,0) to (1,1,1). The work done by F along C, where C is any pws curve joining (0,0,0) to (1,1,1) is integraltext C F·dr = integraltext C ?f ·dr = f(1,1,1)?f(0,0,0) = 5. 2.(20pts) A region ? is bounded by the curves y = x and y = x2. Consider the vector f...