Let \(\mathbf{F}(x, y, z)=\left\langle x^{-1} z, y^{-1} z, \ln (x y)ightangle\).
(a) Verify that \(\mathbf{F}=abla f\), where \(f(x, y, z)=z \ln (x y)\).
(b) Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\), where \(\mathbf{r}(t)=\left\langle e^{t}, e^{2 t}, t^{2}ightangle\) for \(1 \leq t \leq 3\).
(c) Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) for any path \(C\) from \(P=\left(\frac{1}{2}, 4,2ight)\) 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?