Question

For cand D Let C be the path r(t) = cos(t) it Since)j + 2 hk with Ost.<2t. Compute Se Fdx d) Let c, be the path along the square from (5,5,5) to (7,5,5) to (7,7,5) to (5.75) to 15,5,5) Compute G xpit nowad Let Fl

Answer 1

+2) it (aze xyz t xyz Q. Let Feasy, z) = (yze 3) } caye type det c be the path is (t) = cost êt sint f Ostser. Compute {p.dk + t २॥ with Sola f. dr Fact) • 9' (t) at a Fl 5(t)) = F (x [t) y(),2 (t)) Fl h (t)) = (sint sint & ut sint Cost.it 217 e sint Costit/287) + Costxte 271 + 3 शण sint costat Sint Cost e "74) k. oro + 2 TT v' (t)- sint it cost line integral is sint lost it sin²txt e 2T (f. dr = St-sintet Sind cost to 2 sint of cost, te 2T 20 + 3 Cost + Sint Cost ye sint cost t 2 शा + 4x 21 sintot Esce + سمی۱۱ Sin2t't YT Cosztitt 3 Cost-2 sint Sin2t 41 2T + *) dt 2T sinatt 4 IT ( single [ sinat UT + tloszt 20 + 3 Cost- 2 Sint + 2 dt *3)* CS Scanned with CamScanner

RT 20 sinzt st чл sinat YT + t Coszt Coszt at t ((3 cost-zsut + 1 sat 0 Let Sinatt 4T u Sinatt 411 t Coszt dt = du 2T u 3 sint + 2Cost + 2t e du 2. **(****)-(0+2) 2 T Sintet YTI e (1 -1) + 6-2 4 f. dx 4 Ano (d Let C, be the path along the square from (5,5,5) to (7,5,5) ato (7,7,5) ito (5,7, 5) to (5,5,5) sol. - Parametrization of line (26, yo, 2o) ato (21, 9, ?1) ais given by (t)= (1-t) (xo, Yo, 70) + + ( 34,4993) ostel CS Scanned with CamScanner

D(5,5,5) to (7, 5:5) G' 9, (t) = (-+) (555) + +17,5,5) 7, (u) = (5+21,5.5) 05 ri' (t) (2,0,0) Sf. dr IF(<Ct) ) 57 (4) dt C' G' 2515+20) + 2).2 + 0 +0 ) et der = ol e ses aller Il 2 QS (125+ sot) 4 2) at 5 e II (125+50t) e 50 175 e +2 2 e ci ® (7,5,5) to (7, 35) -G" r2(t) = (1-t) 35.5) + +(335) 7, 5+24,5) 9' (t) = (0,2,0) S. F. dea s F($@7) 97' (t) dt . 82 (t) = = C, C," CS Scanned with CamScanner

2 ( 515+2d) 35(5+2) dt + 3 3 ) 0 ſ 35(5+2it) 15th lot) e 73 3) at I Siles (425tho 35(stat) Q5+10+) e 35x2 35(5+2t) jo e 35x243512 + But 35x7 2 10 e (350 3585 25e 35x2 3882 3547 3 35x2x35x2 35x5 + lo 35X35X4 . Feb Similarly (3,7,5) -> (5,7,5); (1) (1-t) (5, 7,5) ++(5,7,5) 3; () (7-2t; 7,5) (5,75) to (5,5,5) Sy(t) = (1-t) (5, 7,5) + *(5,5,5) try(t) = (5, 7-24,5) similarly evaluate over en Wand sult) and (f. de S F(Sub) nu ett (t) dtf G' and = Figye) Sprcho det (planou v je It G'"' CS Scanned with CamScanner