Problem 2. In this problem, we aim to reconstruct the potential of a gradient field from the given vector field by integration. If a vector field F = (P, Q) is conservative in an open domain D C R?, then there exists a smooth scalar function f (z,y) such that F=V f (in this context. the potential energy {/ is J)- The potential function can be derived through a line integral, a fundamental result in multivariable calculus. a) Choose a point (2o, Y0) D and define J(z,y) as follows for any arbitrary (z, y)e D: (x,) flz,y)= F -dr. {*o, ) Since F is conservative, the int;gtal is independent of the strate that: f,(z,y) = P(z,y) and f,(z, )=Q(z,y). Hint: Take the integ ration path r(t) from (z,, Yo) to*(z where path of integration. Demon. 1Y) as the combination ri+r; r(t)=(t,y), te [zo, ], ry(t) = (z,t), te [vo. 4. b) Using the results of part (a), find a potential function for the conservative field: F= (222 20y). Calculate the work done by F in moving a mass from (=1,0) to (1,0) along the path rt)=(t,21),te [-1,1]