Work done by a force field F=yzi+zxj+xyk

F = yz i + zx j + xy k. Notice that F = ∇(xyz): ∂(xyz)/∂x = yz, ∂(xyz)/∂y = xz, ∂(xyz)/∂z = xy. So F is conservative, curl F = 0, and the work depends only on the end points.

φ = xyz
W = ∫_C F·dr = φ(3,3,2) − φ(1,1,1) = 18 − 1 = 17

Direct check along the line: r = (1 + 2t, 1 + 2t, 1 + t), 0 ≤ t ≤ 1, dr = (2, 2, 1) dt.

F·dr = [yz·2 + zx·2 + xy·1] dt
     = [2(1+2t)(1+t) + 2(1+t)(1+2t) + (1+2t)²] dt
     = [4(1 + 3t + 2t²) + (1 + 4t + 4t²)] dt
     = (5 + 16t + 12t²) dt
W = ∫₀¹ (5 + 16t + 12t²) dt = 5 + 8 + 4 = 17

Answer: Work done = 17 units.