Conservative forces are defined as the force for which,

Define conservative forces. What are the properties of conservative forces?

In a gravitational field,if the work done along paths $1, 2$ and $3$ is ${W_1}, {W_2}$ and ${W_3}$ respectively,then:

Given below are two statements:
Statement $I$: An object moves from position $r_{1}$ to position $r_{2}$ under a conservative force field $\vec{F}$. The work done by the force is $W = -\int_{r_{1}}^{r_{2}} \vec{F} \cdot d\vec{r}$.
Statement $II$: Any object moving from one location to another location can follow an infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
In the light of the above statements, choose the correct answer from the options given below:

State the conditions for conservative forces.

The potential energy of a certain particle is given by $U = \frac{1}{2}\,(x^2 - z^2)$. The force on it is: