Question

4:L1-2 Consider the vector field F on Rgiven by () F(x, y) 0 and the curve c: (0,1) + R2 with 2++ + cos(t) Compute the line integral de) = ( Cand cele). \<F1ds>. { <F | do>- <insert a positive integer> Ono For partial credit, fill in the following. You can use sage-syntax, or simply write text. Note that not all ways of solving this problem depend on all fields below. Is the vector field conservative? Oyes If the vector field F is conservative, a potential for F is given by f(x,y) = 1 In terms of a potential ffor F, the line integral is given by Is the curve closed Ono Oyes The speed vector of the curve is (t) The vector field at the points of the curve Fle(t))

Answer 1

Dare Page F(ny) f the given carne us (A) on [0] by 2tt2 3t2 + cost line integral fcilds (An 3î to?) . ( dai fdy ) along curve x= 2+ 12 3624 cost & 1 go aboue integral becomes line integral [(4 (2182) 3 a to to1 )(042642) +dle "fast 0 on line Integral faconse (462482178701):(-284946374.67 $ink)+7) tako dot produel

4 (2+ + 23 2d at 4o. (estist 6t - dint) dut to fucrenys de f 84(8+th + 3-2-42 (2+ + 2)) at 8t (8+ tot 12d2f 644) dt 8 8t+t7 + 12+3+ 6+5) cht 35 (8t+t? 3 [84 + 1264 616 4 utet +8 eye 8 . 8 = 8 32.6 lt 24+ 8 8 65 66 8

Pag for the field F 43 to be Conservative should be zero ie here i = 43 C + + on tok thon cardo 1 С > > y z 3 20- > > A 20 4x3 х sy + - O curd . ( is conservative force fidel. F is conservative 3 potential of sot f If 4 x3 ton Polo = (orio 요 4n3i tog tok i of Ju JE i sf + test 271 y

Date Page on amparing dd > of dy integrate a becomes Of in Ion Ichet gcy) & y respectively gry, is the frofy alme) where f any gly) f X + & lat Joy fody & thix) fa oth (n) fa hon) IV by observing < (v choon 9492 fony We get from X2 to in terms the potential of f the line integral

of ha py + 12 ny ca 1 1309 (4x3 to e 3 to die internal for la CF (ds) (una 1 anitelyî) c line integral - undn as 2fa2 2fo 2 & for tal 2t AC 2-1 3 the integral 4n3ch you com verify this also gives the live integral 65.

Page: curve not closed for lintial 24o pot cogo 2 cw 24 l e3+ cosi 3 etcos) Colt clearly to carve is Not closed # f (6 = 2tt2 e2d2+ cost c' (t) = (2444) tu 22 alle 3th cost) 3 sta sint & fa) 4.(2442) 3 just put ny from thy curve c Feat) 4 (2ff2)3