Question

Show that the differential form in the integral is exact. Then evaluate the integral. (2.0.2) | sin y cosx dx + cos y sin x dy +7 dz (1,0,0) Compute the partial derivatives. OP ON ду az Compute the partial derivatives OM OP dz Compute the partial derivatives. ON dy

Answer 1

(2210,2) siny rosx d2 + cosy sina dy+Fdz (1,010) we let M siny rosa cosy sina -OM ƏN əz. әр 22 az P- 7. OP=0 OM -0 - ON am sco Sycosx- oy Z. ay These equalities tells us that siny Cosz dx + cosy Sinx + dz is exact, so df sing cosa dx + cosy Sin a + d2 = for- some function f, and the integrals value is f (2012) - f (1,0,0) We find f up to a constant by integra- - ting the equations. of ag oy of sina, = cosy sing cost, @ 7 οχ / g is ist from the there equation, we get f(x, y, z) = siny sina + g (y, z) the and equr we get, af - cosy sinat og ay oy oy cosy sina, or og 0

Hence . g is a function of z alone, and frary, 2) siny sina thrz) further molle, Of oc - 1 = 0 + ch dz 07 hiz): 2+ :. f(xyz) - Siny sin 2+ z tc. the value of the line in tegoal is independent of the path taken CQ10,2) to (1,010)and equals f (2,0,2)-f(1100) = 0+2+8.-06. - . El ap oy 0 ON az -0 OM dz =0 OP 02 ON Cosy cosa OX OM oy • cosy cosa