Question

Let F = (P,Q) be the vector field defined by P(x,y) ity, (1, y) = (0,0) 10, (x,y) = (0,0) Q(x, y) = -Ity. (x, y) = (0,0) 10, (x, y) = (0,0). (a) (3 points) Show that F is a gradient vector field in RP \ {y = 0}. (b) (4 points) Letting D = {2:2020 + y2020 < 1}, compute the line integral Sap P dx +Qdy in the counter-clockwise direction. (c) (1 point) Does your calculation in part (b) violate Green's theorem? Explain.

Answer 1

Solution en Given that TH F = (p, q)

where 17 p(X,Y) S xty x+y2 1 (716) + (0,0) Giy) = (0,0) o and Q(x4) -2+y x²+42 (7,9 +00 o (914) 1010) f @ To reefor show show field is gradient in iR² \ {y=o) o in IR² {y=0) we have to that curl Now curd F = ☆ T3 a Q ax of fi) 29 22 2 aa -a ty nty² When (117) + (0,0) a (+42) (-1)-(-x+y).22 (nty 22 ay² + 2x²_2ny x-y-2ny Fortyn)? (x² + y2)2 x 2P ay when (A) aty 2²+y2 op #100) ay 22_y²_2ny (22742)? Cat42.1-(4+1).2% (11742)2 29 - a P ay

Hence from (1) Curl F = â .o 11 Po ਵੀ is veetor field in a gradient (proved 2 IR 1 {yzos. 2020 8 2020 (6 D = 7 ty LI closed curve. Here D is Now we Know that for veetor over field integral of the equal s (pdxt Q dy) gradient region, the line field along any closed zero. is to curve ܢܐ O. &D

e) theorem stales that Green's S (pdx dxt Qdy) - SS@ 2P) dady әх ay dD Now Since әР 2 Q 22 ay = 0 s (pdx + ady) = sso dady SD which we already proved. part (6) does not The violate calculation in Green's theorem.