Conservative Vector Fields

There are two ways to determine if a vector field $F=(F_1,F_2,F_3)$ is conservative. In the first one, one tries to find by inspection a scalar function $f(x,y,z)$ such that $F(x,y,z) = \nabla f(x,y,z)$.

This is the case, as it is very simple to check that for $f(x,y,z) = -\cos{(x^2+yz)}$, $\nabla f = F$.

If the first attempt proves too difficult, one can always check if curl of $F$ is equal to $0$ and if $F$ is defined on a simply connected region (with no holes in it). If that is the case, the vector field is guaranteed to be conservative.

This is the case, as $\text{curl}(F) = (D_y(F_3)-D_z(F_2),-(D_x(F_3)-D_z(F_1)),D_x(F_2)-D_y(F_1))= (0,0,0)$, and the domain of $F$ is the whole Euclidean space.

To find the potential, we integrate $F_1$ with respect to $x$ and check if said integral is the right potential (this is the case in this problem). If it is not, we add a function $h(y,z)$ to said integral, derive it with respect to  $y$, set it equal to $F_2$ and determine the previous function $h(y,z)$ up to a function $g(z)$. Once again, we check if we have found the right potential. If we haven't, we derive the function from the previous step with respect to $z$, set it equal to the $F_3$ and determine $g(z)$ in this manner.