Verify that \(\mathbf{F}=abla f\) and evaluate the line integral of \(\mathbf{F}\) over the given path.

\(\mathbf{F}(x, y, z)=y e^{z} \mathbf{i}+x e^{z} \mathbf{j}+x y e^{z} \mathbf{k}, \quad f(x, y, z)=x y e^{z} ; \quad \mathbf{r}(t)=\left\langle t^{2}, t^{3}, t-1ightangle\) for \(1 \leq t \leq 2\)