Question

Line Integral & Path Independency Problem 1 Prove that the vector field F = (2x – 3yz?) { +(2 – 3xz) j-6xyzk is the gradient of a scalar function f(x,y,z). Hint: find the curl of F, is it a zero vector? Integrate and find f(x,y,z), called a potential, like from potential energy? Show all your work. Then, use f(x,y,z) to compute the line integral, or work of the force F: Work of F = jf.di from A:(-1, 0, 2) to B:(3,-4,0) along any curve that goes from A to B. First, check if an f exists. تح k curl Ě = XĚ = a a a ах ду дz o] x [Ex Fy F] = a ax a ду a az 12x - 3yz? 2 - 3xz2 -6xyzl Is the work of F path independent?
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Answer 1

Answers F = (2x - 34 22)i + (2-3x22) ĵ - (Gry z) Å L Curl = NxF Z d d dx sy d2 - 6x42 by Curl F SY 2x-3y22 2-3x22 Solving the determinant. 662va)– (3 (2+372)) -Co .(-6442) - (2x - 3422) »2) À * ( ****t) (22-392)] 2123-3422), 3(2x-347) [-6x2 + 6x2] ů -[-6y2 + 6Y ² + [-32+32?) Culf = CurlF = O a B A Thus, Carl is zero vector. *F is the gradient of a scalas function Plex2). wank of E dr. S cont ds = 57. ds (by stokes theorem. Curl=0 * thas, wook of F path independent. So, YES path F = (2x - 34 22); +(2-3222) j - (6x42) i is Independent Fx=2x-3y 2² fy= 2-3x22 f = -6xyz Fx = 22- 3422 da dx = 2x 3Yz²x = x²-3y z²a ) af = f | 2x - 34 2² dx 2 df = Fy dy

f = J Fy dy 12- 2- 3 x 22) dy at f - 2Y - 34x22 + df đz NY - - 6xyz - f = (-bayz dz f 3 xyz². * Thus, by grouping without repeating terms, flax F x² + 2y 3xy 22 ensure - = 2x-3y22 df - - - ) * To all three requirement meet, dt d (x² + 2y - 3x422) Ex da da d (x² + 2y - 32422) 2-3x2² = Fy dy dy df den (x2 + 2y = 324 32)3-6x42 = F dz This at Fx, dt = F₂, df = { * The line integral is, St. dr = 18/3,-4,0) Sr.de x²+2y - 3x422) x2 +24 – 3x42²) (3,-4,0)=B st du = (9-8-o) - (1+0.0 so . d = 0 Thus, the line integral is B f (x, 4, 2)) А 46,0,2) 11.10 +(-1,0, 2) =A