Question

Chapter 15, Section 15.3. Question 015 Confirm that the force field F is conservative in some open connected region containing the points Pand and then find the work done by the force field on a partide moving along an arbitrary smooth curve in the region from Pto Fix,y) = xy + y): 7.9.068,0) Enter the exact answer as an improper fraction, if necessary W- Edit

Answer 1

F(x, y) = xyz + xy Ca Curl, F = VXF = ole ele o :00 L oy xy? e²y -1860-2-9]+12.03-90] ti [aze (2*9)-3(293)] - oitoft [2xy + 2xy] That is Curl F = 0. Hence, the force field F is conservative. Then there exist potential function $(2;4) such that F(x,y) il, xy² +2²y ربا രg ره - 00 oy OR Then rol = 20 innya and og = rezy rore oy Now da = roe du + oody x y 2 de it e2 y dy

= (2xy dx + 1x'y dy] į d (oz?y?) lie, do dº = d ( 22 2 inte grating we get 0 (²9) = 1/ x²y2 +4, where ca is the integrating constants -! arbitrary The work done by the force field on a particle moving along an Smooth Curve in the region from P to g, is given by W = fF.dr C с Since the force field F is conserieative, the integrat should be path independent, but depends and on the realues of ø at the P and o. points The work done w ff.dr = $($) – PCP) => W=0(8,0) – 0(7,0) = , - 3959 Ci W = 3969 2