Apparently a gradient field has special properties that not all vector fields have one being that it is conservative. However the gradient field only exists if there is an exact differential and from what I understand this is a sufficient and necessary condition.
Is there property of a differential that is the same as conservatism? ( i.e. A differential is a place holder for the addition of partial derivatives and the gradient vector field is that addition resulting in a vector that points in the direction of maximum increase of the slope. It is not surprising that it is the direction of maximum increase, since if you skew it as can easily be done by a directional derivative away from the original basis vectors the Pythagorean theorem shows the rise over run will not be as steep.