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, V f = F) with f(0,0) = 0. If it is not conservative, type N. A. F(x, y) = (-6x - y)i + (-x + 14y)j f (x, y) = B. F(x, y) = -3yi – 2xj f (x, y) = C. F(x, y) = (-3 sin y)i + (-2y – 3x cos y)j f (x, y) = Note: Your answers should be either expressions of x and y (e.g. "3xy + 2y"), or the letter "N"
$(1 point) The domain of f(x, y) is the xy-plane, and values off are given in the table below. \y\x0|1 2 3 4 0 50 50 50 50 51$

Answer 1

an A F(2,4)= (-6a-yi + (-x+144); = piti Where P=-62-y & g = -at144 29 ay da De 4+1=0 - F (ay) is conservative field J a scalar pottential fray) st. If - F & df-fodr indf-Fodt= (ba-y dot(atlaydy indf df = -bada-yda-ady +144 dy =-bada +14ydy - Cydatady) dfa badatlaydy-dlya) > f = -69 + 14 y yate --- -3 + 74 yate f(0) = 0 c=0 =-32+74-ya Indf 2

24 there on B) Fay=-347-22] =pi +3j Where p=-3y , g=-22 ag - OP = -2-6-3) =~2+3 =1 to > Fis not conservative field (2) F (a, y) = (-3sing) i + (-24-32 cosy - Pitsi p=-3siny f92-2y— 3a cosy og -op = -300sy - (-3cosy) dy Fis consentative field. - a seatar potential of st. & df=pf odr Fodr indf--3sing dat (-24-3acosydy - f = S-3 sinyda ot SE24-32 Cosy)dy to y-const f =-asiny + faydy to Basiny - 27 -Basimy-yte foo=0 (flat) zasing TL = free from 2 ezte 2 =0 /C=0 = -30