M. C. Escher’s painting Ascending and Descending illustrates a non-conservative vector field, impossibly made…. In reality, the height above the ground is a scalar potential field [the scalar (single number attached to a point) being the height above the ground]. If one returns to the same horizontal place, one has gone up exactly as much as one goes down.
So that’s that picture related to
Conservative vector fields obey the product rule:
conservative scalar field is also the output of a derivative operation…just a different dimensionality)