Final answer:
To show a vector field is conservative, it must have a zero curl or a potential function from which it can be derived. For electrostatic fields, this implies a path-independent work integral, contrary to magnetic fields which are non-conservative and governed by Ampère's law.
Step-by-step explanation:
To show that a vector field F is conservative, one must demonstrate that the line integral of F along any path from point A to point B is independent of the path taken. This can be achieved if the curl of F is zero or if there exists a potential function U such that grad U = F. Additionally, in the context of a conservative vector field, the work done when moving along a closed path is zero.
In the provided physics example, we are given that for a two-dimensional, conservative vector field, the condition (dFx/dy) = (dFy/dx) holds, which implies that the partial derivatives of the components of F are equal and hence the curl of F in two dimensions is zero. To find the magnitude of the force at the point where x = y = 1m, one can directly use the formula provided where the partial derivatives equate to 4 N/m^3, leading to a straightforward calculation of the force magnitude at the given point.
It's important to remember that electrostatic fields are conservative, meaning that electric forces are also conservative. However, magnetic fields are an example of a non-conservative vector field, as they do not satisfy the condition of path independence for their line integrals and are instead described by Ampère's law.