Gradient vector over an area of a surface

[trancefishy]

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?

 

[HallsofIvy]

You say "the gradient vector on a 3d surface" but I think you mean the gradient of the function defining the surface. I.e.- f(x,y,z)= constant so you can think of the surface as a "level surface" of f(x,y,z). If that is what you mean then grad f is always normal to the surface. Further, its length, times dA, is the "differential of area" and so grad f itself can be thought of as the "vector differential of area".

 

[trancefishy]

nope, not really what i was going at at all.

state differently, perhaps more simply, is I want to know how to find the equivalent of a gradient vector of a surface f(x,y) over an arbitrary area, not just a specific point.

now that i think of it a little more, this seems a bit absurd. an analogy is, say i have a garden that's a bit uneven and it's raining. normally the opposite direction of the gradient(at the point I'm standing) woudl tell me which direction the water would flow away from my feet. what i want to know, is taking the entire area of the garden into account, what are all the different rates of flow, max, min, etc for the whole area. we are assuming that the function defining the surface of my garden is continuous and differentiable. etc.

this is beginning to sound like a vector field now...

 

[HallsofIvy]

nope, not really what i was going at at all.

state differently, perhaps more simply, is I want to know how to find the equivalent of a gradient vector of a surface f(x,y) over an arbitrary area, not just a specific point.

now that i think of it a little more, this seems a bit absurd. an analogy is, say i have a garden that's a bit uneven and it's raining. normally the opposite direction of the gradient(at the point I'm standing) woudl tell me which direction the water would flow away from my feet. what i want to know, is taking the entire area of the garden into account, what are all the different rates of flow, max, min, etc for the whole area. we are assuming that the function defining the surface of my garden is continuous and differentiable. etc.

this is beginning to sound like a vector field now...

That's exactly what it sounds like! And, therefore, equivalent to a differential equation where the gradient itself is a derivative.