Question

(a) Given the vector field F = (0,22 + 2xy) = ui + (x2 + 2xy)j Find u for 7 to be conservative and find the potential, if it exists (b) Given u= (e? – zły, xy + y) = (e– r’y)i + (xy + y); Evaluate I= dos u dr where is the circle with radius r = 1 and center at the origin.

Answer 1

Answer a given field F = xu, 2²-gay> = uit (a?tday) i ty F is conservatia than OXF zo ol 1 hele vele y %oz azazy o → Frosth10) + (2x+2y-) he = Qatay he constant, = Qay+y²c , cis {:0=1% at %% et its potential it F = vt (Qxy-tytc) it (xit say) = ? at a to at ten Re IT of TT. = gay+420 an B and of Prester - dy from equ @ te f = x²y + xy²+(2+0(4) honey x²+ qaytag from egn he and 5 $=k 50 Prestupe = 30 x² + qayt a dy sy where k is integral Constant

so from egr potential of ř is f = xy + xy?+ cae+k 4=(62_a²y) î t cay²ty3] Ic=x²+4²1 To $4.67 >I= fi {(exxy ) î+(ay'ty ) }. (daî tayh) ge?_xy) du + (xy? + 8)dy - x74²1 = can use inreen theorem if cis where ter c is closed pathe R therefore we tra - according to green theorem closch path and I= de Moda Indy R is enclosed by curre IN I= IN sy then from egna I= Sle (y + 2 (2) andy ken R: x² + y²1 put aa ruso, u= rsino didy = r drdo = x²+u²r² r²1 =r=1 TT QT so, s p?ror do I= s 030 720

do Voo ( I do 18 나 [] 1) 1 서g there fose T: 7 Answer.