Question

Determine the potential for the field: F = (2y2 + 20e4x-2y, 4xy + 5 - 10e4x-2y) Do not put the constant "+c" for the potential in your answer below. f (x, y) = Submit Answer Tries 0/8 Now calculate Si F. dr where C is the path ř (t) = (2t, 4t) for 0 <t<1 The line integral equals

Answer 1

Solution : F = {zy? + 20e42-24 , 4 xy +5 -10**-27) Here, Fi 2 y² + 20 e 42-24 F2 = 4224 4xy + 5-10 2 NOW OF dy = 4 y 40 e 4x e-zy a F2 4 * e-22 11 40 ty to Here, 2 F2 OY Curl of F is zero. afi ay Hence, vector vield is Conservative Now, df (x, y) (o) - (F1, Fe) = ( 2y*+ 20e ***** ****5-10****y) df (x, y) = 2y² + 20 e 4x-2y a f(my) 2y2 + 20e-24 4x an 4 x f (x4) 2 y ² x + 20e + g(y) 4 43-24 f (x,y) = 2y²x + 5 e + g(y) (3)

jnow, Differentiating above eqh wrt y af (**) dy =qxy + 424 5 e e-24 (-2) + dg(y) dy d f(x,y) 42-24 4 ry - 10 e + dg(4) dy we get Comparing this ear with Ear @ daly) 5 dy gy) = 5 y tk 3 Combining this ean with 42-2y f (ve, y) = 2 x y² + 5 е + 5y + x Hence, Neglecting constant k 420-2y f (x, y) = 2x y² + 5e +5y we get Now, SFide Fax + F₂ dy و کا с 1 y = 4 + we have r = (x, y) iso, X = 2 + we put everthing in terms of t'

da = 2 dt dy = 4 at 2 2 2 .: Fi 8t - XL 2 (4+) + 20 e 32 t + 20 F2 4 (2+) (4+) + 5 - 10 e -> Fi = 32+² +5-10 32 +²5 in Now can put all of these the integral since t runs from o to 1, Cour limits) I- Adx + Fz dy = I= → fore'ora) a ne u frees) au La force 1-2 foca'10) 2[132 + + 64+*+20 -10) de I 11 → I - 1 84 >> I 2 [ 32 + + 10+] SEdo Hence, I. = 84