MATH 317 PROBLEM SET III SOLUTIONS1)Determine whether or not each of the following vector fields are conservative. Find the potential if it is.a)~F(x, y, z) =xˆı-2yˆ|+ 3zˆkb)~F(x, y) =xˆı-yˆ|x2+y2c)~F(x, y) =yˆı-xˆ|x2+y22-y2+32)is conservative.Solution.c)~F(x, y) =r(tan-1xy2)Let~Fbe a conservative force field with potentialφ. Show that the lines of force are perpendicular to theequipotential surfaces.3)Find the velocity field for a two dimensional incompressible fluid when there is a point source of strengthmat the origin. That is, fluid is emitted from the origin at area rate 2⇡mcm2/sec. Show that this velocityfield is conservative and find its potential.