Question

8Two vector fields are given: F(x,y,z) - (esin(yz), ze* cos(yz), ye* cos(yz)) and F(x,y,z) = (z cos y, xz sin y, x cos y). a) Determine which vector field above is conservative. Justify. Foly = fjol so, <ea sin(J2), 20% cos(82), y acos (92)) Conservative. b) For the vector field that is conservative, find a function f such that F - Vf. Lxelsing2, zetos yea, yet cosy 2 c) Use the Fundamental Theorem of Line Integrals to find the work done by the conservative field as a particle moves along the curve given by r(t)-(7.7 + 1,7% +2,05152.

Answer 1

solution: tenis @ F ($9, 3) - Kem sin lys), pe" cos ( 43) , ye?cos (93)? Carl Fri ; 6 Carl F F by le "Sin (Ys , &e Cos (43) Ye Cos (73) 27 ("coH4) - y3e"sintys) - e' cor 43) + ube 58145)) -j ( certya) - ye Sereys. Itk le sotu 4) - ze" Cos cyt) ) ico,o, of recla field. Flence fis Conservative $ Consider ri c os(4), x3 sin (Y), A Cos (4)7 cool if a 2 Coly) 13 Sincy) AGA (4) z (-X Sin (4) - X Sin (4) J ( Cos (Y) - Cos (Y)) + te (& finly) + & Sin (4) a {-2x sin(4), o, 23 Sin (4) 7. ence, corl of to les & is not conservative vector field in this case.

Lot - of of delay - from 0 & we get of een sin (43) ofo = B e Cot (94) * Lo at yeu cos (43) fase sinya) du Ifa en sin (43 +9 (9,8) det en Cos (Yå) + 9 (4,8 g from ② x 5 ze Cost y ) + 9' (93) = ze "Coltys) g' (9,8) :0 919,3) = h (3) fa en Sin (Y3) + h (3) ose eyes Coll93) + 'CH) from @ +6 yea Cost y eľ (3) 2 ye? Coslys) h'(alco h(a) z Conetant (6) Hence, fi ea sin (43) + k

o octez &(t) = {th, th41, +²12) (810) 2 (0, 1, 2) 7 (2) 244 4+1, 4+27 17(2) i L4, 5,67 By Fundamental work done theorem ff.dra f ( 0(2)) - flr(0)) if <4,5,67 – f (0,1,27 c ell sin (30) - Sin @)