1. Show that the integral: ?(1,1,1) to (2,2,2) (x + 2y + 5z) dx + (2x - y + 3z) dy + (5x + 3y - 2z) dz is independent of path, and evaluate it. 2. Evaluate the work done by the force: F(x, y, z) = sin(x²) i + cos(y²) j + z²k, on a particle moved along the path: r(t) = cos(2t) i + sin(2t) j + 4k, 0 ? t ? ?.
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First, we need to check if the given vector field is conservative. A vector field is conservative if its curl is zero. The curl of a vector field F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k is given by: Curl(F) = (∇ × F) = ( (∂R/∂y - ∂Q/∂z)i - (∂R/∂x - Show more…
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