Let F(x, y, z) = (xz, yz, ln(xy)). (a) Verify that F = Vf, where f(x,...

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Let F(x, y, z) = (x−¹z, y−¹z, ln(xy)). (a) Verify that F = Vf, where f(x, y, z) = z ln(xy). Se F. dr, where r(t) = (et, e²t, t²) for 1 ≤ t ≤ 3. (b) Evaluate (c) Evaluate Sc F. dr for any path C from P = (1,4, 2) to Q = (2, 2, 3) contained in the region x > 0, y > 0. (d) Why is it necessary to specify that the path lies in the region where x and y are positive? verify that F = Vf and evaluate the line integral of F over the given path. Let F(x, y, z) = (x−¹z, y−¹z, ln(xy)). (a) Verify that F = Vf, where f(x, y, z) = z ln(xy). Se F. dr, where r(t) = (et, e²t, t²) for 1 ≤ t ≤ 3. (b) Evaluate (c) Evaluate Sc F. dr for any path C from P = (1,4, 2) to Q = (2, 2, 3) contained in the region x > 0, y > 0. (d) Why is it necessary to specify that the path lies in the region where x and y are positive? verify that F = Vf and evaluate the line integral of F over the given path.