dentify the nature of the field, if the divergence is zero and curl is also zero

If a vector field has both zero divergence and zero curl, this means it is a conservative field or a potential field. In physics, this typically represents a situation where there are no sources or sinks in the field (zero divergence) and no circular or rotational motion within the field (zero curl). An example of this would be the electric field surrounding a set of electric charges in static equilibrium. In such a field, the work done to move a test charge from one point to another is independent of the path taken between the two points, and there is a scalar potential function from which the field can be derived.