Question

Find the work done by the force field F(x, y,2)= <2ay - :, x° +23, 2y-2x > in moving an object from point A(-3,-2,-1) to point B(1,2,3) along the following paths: a line segment followed by the arch of a cycloid, followed by the top half of a parabola, and followed by another line segment at the end. Evaluate for full credit. (9 pts)

Answer 1

solution: Slepol; Given F(X, Y, 3) = (any - 3? x²+28, ay-203) A (-3, -2,-1) & B (1,2,3) -- k 0 Now Curl F : d d OX d ay any - 32 x+28 ay-ax} (2-2) -j (-23 +28) tk (1-A) {o,o, o) we Hence, given vecter feld F is Conservative know that for the poléntical function F: of =) Ray - 32 = of a? + +23c df of dy de fe Q2%y 3x +919, 8) & 24 2x } E f: x²y - 32+ g (4,3) df a ? + g'14,3) Ч from © & ③ we *+29 = 24 + g (4,8) 22- g'(4,2) 24 & thea)= 919, 8) teace, fi x²y - 22 x + 243th (3)

A 2x3 +24th (3) from we ay - 2x3 = - 063 + xy +h' 13) (2) 20 hladec Hence, fi azy 3²x+ 243 + conelant (c) grow work done i Sf.de C (1,2,3) C d (x’y -32x4 24 3 + c) (-3,-2,-1) 2 (1,2,3) x + 246+ ( ) (-3, -2,-1) -[11*ca) - 133220) + 2 (2(3)+c] - [4337- 2) (-1)+(-3) +26-2)(-1)+5] ( 2- 9+12] - [-16+3+4) : 5+1 16