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) = (-14x + 4y)i + (4x + 2y)j f (x, y) = B. F(x, y) = -7yi – 6xj f (x, y) = C. F(x, y) = (-7 sin y)i + (8y – 7x 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"

Answer 1

. A. Given F(x,y) (-142 + 4y)i +(42+29); find the Curl of the with it the Curl of the water is zero then it is conservative otherwise not conseruative i j VXF 2 읊 ay -ja e -147+4Y 42+27 42+24 0 K 6 a . 2 az c 0 ar ay az 92 +K -1474 0 -147+47 42+24 11 (309-36-1-7)+ Je CE (0– (42+27)) i @-0) - j (0 - 0) +(1434 o) – i (6) - 3 (0) + K (f+0 - 0 -*) feat)(120) (-14(0) +40) 1 = (2) 11 (O) = 0, 0-0 +0 0 0 = 3XA | IZ af 21 ау -142 +47 of JC re 32 so it is conseruatie . Then fend potential fumition f of - i of + j of +* Of 4x+2y integrating with respect to y Prize,y) -> (23) +428 + 424 + (3) 0 +4:2 (1) +968) 45+24 4€ + 3'6) → 5'(4) = 24 on (0 <-- (9)6thäxht_XL- (6) 6+ach Oy

integrating with respect to y Sos) = a[y di 9(Y) - ay El from (1) f(z,y) -7.20² + 4xy + y2 2 16,4272+424+19) b) Fla, 4) =-7yi - 6xj VXF e j 2 ay o o lot aze az - 74 -62 cila - 2 662) ay 462))-;( (0)-267): fatel az O =K 0 (0) - i (0-0) -; 6-0)+* (-60) - 670) -0 + (6 +7) VXF Not Conservative so N. :. f (,y) N c. F(x,y) fisiny)i + By-72 coby); VXF a aze ay |-7 siny 8Y-126by 0 xFE 20-2 (BY- j k a eldo j +K (89-1969) (Sing) i(0-0) -3 6-0) +K (2:0) - 768 y(1) +7 (468Y)) 0 - 0 + (- TC87 +1Ger) 2 20-00 VXF= 0

so it is conservative then find potential function f. If ľ af az + j af ay tk af az af -7 siny → (2) 22 84 - 7a Cosy (3) of ay af 22 =0

(9) = from (3) = -72 siny +964) 야 -7ze coby + g(y) ay 84-72 Cost = -72664 +34) g'(9) sy vñtegrating with respect to y ၂၁(၂)dy = 8 Sydy 9(4) = 8 y? 472 2 Substitute 9(4) in Equation (4) 59) -72 Siny + 4y2 f(e,