what follows is a question I asked myself, the answer I figured out, and the new question that arose as a result. I was thinking about the gradient vector on a 3d surface, and how it shows the direction of the max rate of change at a point. the 2 directions perpendicular to it are tangent to the level curve thus represent zero change, and the direction opposite the gradient represents the minimum rate of change. so i was trying to think of a surface where this was not true. say, a tilted plane, where, halfway between the origninal min r.o.c. and the zero r.o.c. direction, there was a divet than ran "down". this would then be the min r.o.c. and certainly not perpendicular to the zero r.o.c. direction. I concluded that this surface would not be a continuous function at the point in question, due to the divet needing to stop abruptly. i had to remind myself that i was talking about a point, not a very small area. so, how would one compute these gradients over an arbitrary area on a surface? would this be extendable to a curve that ran along the surface?