Question

1. (20 points) Identify if the following vector fields are conservative. If there exists a vector field that is conservative, you must also find a potential function for that field. (a) F(x,y,z) = (x3 – xy +z)i + 2 (b) F(x,y,z) = (y+z)i + (x+z)j + (x+y)k (& +y +y-22) i + (- y2)k

Answer 1

solution: step 1: F(x, y, 3) (2 ny + 3) it ( 2 + y - 235 + (x2-43) k а ke i لی d on dy a3_X YtZ 인 [cwe know that if Carlo io. then 'f' is Conservative vector field Cul प ah + y - 23 x2y3 ci (-3 +2) - j (2x - 1) + T ceny - ~) (2-8, 1-02, 0) Cualfa <2-3, 1-an,o) to Thus, f is not a conservative vector field. © f(1, 4, 3) < (9+ 3) 7 + (NAB) + (a tył Girlf e 2 4 Ja ata Hence, Cuale A TE -- E d oy Ja xty y e3 +7(1-1)-j (1-1)+F(1-1) i {o,o, oy

co - ) da dy Hence, Curlf ches f is a conserwative vector feld Now in the Case we can find potential function. for potential function faol of df Fil If ) From Q & we get df of Cat & of dx = 9+8, fe ay+23+ 914,3) since integrale df g'(4,3). 5 from ② k ③ we get : at 9' (Y, 8) = x + 3 g'(4,3) = gcy, a) = 48th (3), since integrate on both sides Thes, fa ay+2 3+4 34 h (3) df Jy - aty. dfa y{}. du ( = xt dy ® of = x+yth'cs) from @ < 6 we get x+4+h'(a)= x+y R (&) = 0 =)/ h lalec potential Hence, f = ny +43 +4 hec the required function is