1.
Consider the force field \textbf{F}(x, y, z) = (2xyz^2, x^2z^2 + z \cos yz, 2x^2yz + y \cos yz). Show that the integral $\int_C \textbf{F} \cdot d\textbf{r}$ is independent of the path. Then use the fundamental theorem of line integrals to calculate the work done by \textbf{F} on an object moving along a curve C from (0, 0, 1) to $(1, \frac{\pi}{4}, 2)$.