Question

(1 point) For each of the following vector fields F decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, Vf = F) with f(0,0) = 0. If it is not conservative, type N. A. F(, y) = (12x - 4y)i + ( 4x + 14y)j f (1,y) = B.FI,y) = 6yi + 7xj f (, y) = (6 sin y)i + (-8y + 6.0 cos y).j C.F (2,y) f(x,y) — Note: Your answers should be either expressions of x and y (e.g. "3xy + 2y'), or the letter "N"

Answer 1

for conservative force, VỀ so FR2 ax a ay 22 A Post Haz 3 x loy (122-44) (-4x+144) 23 (-4x+184) -1 [o- (2x - 773] + (=9x+143) – 2 (128 - 4437 NE=4-(-4)] ау txo They ř is conservative 9. F Since J x can be written as guadient of any function of F = 7 f where di i de + jdy + Edz 12a-44) i + (- 4x4 (44) (ty) Antj dy tkdz}

This field is not conservative. 2 f C) Zey 1 EXĚ ron Hay /02 Gsiny toý +62ecosy) + 2-02-10-6] + 21-04+ 6xcosy) 2 (osing) 2 ay Ky 5 Eo conservative Geasy - Glosy field Ě is Now f = ff.de +4-8y +fxcosy); }, {idx=jdy the f = fflesing) ? 4 esing da + ffady + brcosy) dy

By2 + bxsing f = 1 (6.sing)? - 84² + c 2 Ce constant of inte integration at Cool f(0,0) o 0 o tofc floo) C= f(0, 0 = 0 f(x,y) = 6xsing teasing - 942 f(y) = x6 siny = 442 f(x,y) = 12 sing 2 44²
Summary: for a conservative field curl of field should be zero.