Math question about the path formed by following a gradient

What is the name of the line (or curve) that connects the points corresponding to the maximum (or minimum) values of the gradient.

I doubt that’s clear, so let me give two examples:

  1. If you are atop a mountain (assume there is a unique ‘top’ point), and you proceed to go down the mountain by always moving in the direction of maximum slope, is there a name for the path you wind up taking? (assume the surface of the mountain is everywhere smooth, continuous, differentiable, etc.).

  2. Imagine you’re in a pool of water where the water temperature at each point varies (and, again, assume continuity, differentiability, etc.). If you always swim in the direction of the hottest ‘nearby’ temperature, what is the name of the path or curve that you’d take? (As an aside, am I correct in assuming that this path or curve is the same as if you were always swimming in the direction of the maximum ‘slope’ of the temperature, i.e. where the temperature change at any point was greatest?)



Not that I can say it with complete confidence, but I think geodesics are a related notion which is used to describe the shortest path connecting a set of points but without regard to (what I am calling) the gradient.

Note that the path isn’t necessarily unique, and that when constructing such a path (always following maximum local gradient), I suspect that you can get a different path starting from the top than starting from the bottom.

However, there will be a unique set of shortest paths, which will have the greatest average gradient. They just won’t always follow local maximum gradients.

(Note: I’m thinking intuitively and could be wrong, but I can imagine examples.)

As Karl Gauss, I’m sure it’d be trivial for you to prove that the shortest path(s) would have the maximum average gradient. I’m no genius and I think I could tackle it!

Are you talking about the gradient flow?

Yes, in a typical real landscape, the path downhill will follow river valleys, while the path uphill will follow mountain ridges, if it’s maximising the immediate gradient rather than the average gradient.

I’m honestly not sure.

What I’m asking about is probably not even all that complicated - the problem is that I’m likely not explaining it well.

Basically - is there a term for the path (i.e. the continuous curve) that joins all the ‘successive’ maximum gradient points from a given starting point?

For your second one, it depends on how you’re defining “nearby”, but assuming that that means “within some small distance, in the limit where the distance goes to zero”, then yes, that is exactly the maximum gradient.

I think your explanation is plenty clear. I can’t say whether there is such a term. But I will point out that even if the surface function is well-behaved, there’s no guarantee the path is *a *path. It’s more like a tree of paths.

Imagine a surface which is an inverted parabola of rotation. From the peak, the gradient is the same whether you start to descend going North or South or East or any direction at all. so really the entire surface is covered with almost parallel paths that run from the peak down to the bottom. And each path has the same gradient at any given distance from the peak.

Admittedly that’s a pretty pathological case. But an idealized 4-sided pyramid will have 4 paths from the peak; each leading down the center of a face. etc., for many other regular surfaces / functions.

For something irregular like a mountain, even if we imagine away the fractal issues of ever-finer scale, we’ll see lots of points on the way down where there is more than one direction with the same maximal slope. And at each such point the path must branch.
To be sure you could constrain your definition to surfaces where the maximum gradient is uniquely definable at every point or at least every point on the maximum path from your origin.

But that restriction seems to my unsophisticated taste to be pretty much artificial. IOW, if the idea is only applicable in highly constrained environments, it doesn’t seem like a fertile idea for further analysis.

Thanks for your reponses.

It may be that what I’m trying to label is nothing more than the directional derivative (or is a function of it), but I simply don’t know enough math to be sure.

Nah, I don’t think so: the directional derivative depends on which direction you choose to go in, and is greatest if you go in the direction of the gradient.

I think it would be called the riverbed. :slight_smile:
Water flows in the most downhill direction(s)

Basically, I imagined the same problem when standing at the top of a ski hill.
The most downhill route sometimes takes you into the trees, but if you ski more across the hill rather than directly down, you are to some extent fighting gravity.

However, imagine being on the top of a mountain with a parabolic profile. There is no FINITE set of paths since the function is dense(?), for every pair of valid paths there is a path between them.

This is exactly what they’re talking about, whether they know it or not.

(At least, this is 1). Which is the same as 2) at all points where the gradient is nonzero. Where the gradient is zero or not defined, 1) and 2) can come apart, and 2) may also be ill-defined.)

The gradient (of height with respect to horizontal position) will be zero at the peak of the inverted parabola, and thus following the gradient (in the sense of the OP’s 1)) is as good as making no move at all.

Though, the OP had us assume smoothness, so the gradient will be everywhere defined; the only potential issue is points where the gradient hits zero. At such points, the slope in every direction will be accordingly zero, and thus there will be no particular direction of maximum slope.

I’m going to stick my neck in this noose here and comment what I think, and it’s not pretty. The word “gradient” is more commonly used as an adjective, it describes a specific quality of whatever noun it is modifying. Indistinguishable mentioned “gradient flow” which means we have a flow that behaves as a gradient.

In the case of a regular pyramid, and we follow a straight line down, then for every ten feet we travel, we will drop down the grade the same distance. I doesn’t matter where we are on the grade, the slope is the same. This is an example of a non-gradient path. Whereas on the inverted paraboloid, each successive ten feet of distance we travel, we will drop down faster than the previous ten feet. The slope is increasing the further we travel. This is the example of a path that behaves as a gradient.

A gradient describes a specific way something changes, the rate of decent on a paraboliod increases, the increase of water pressure with depth, your dog’s speed when the hamburger hits the floor. I don’t think there’s a specific name for this, other than what names we give otherwise, so in the second example above, we’d call this gradient path a parabola.

“Gradient” in math has a specific meaning which is different from the common parlance way you used it just above.

In math and simplifying a bit, “gradient” means the specific direction from a particular point which has the greatest slope. Plus the magnitude of that slope. Each point on the surface has potentially a different magnitude pointing in a different direction.

The math “gradient” says exactly zero about how the magnitude of the slope may change over distance.

watchwolf49, you might want to educate yourself on the terms being used before you chime in. The word “gradient”, used as a noun, has a very precise definition in mathematics, and everyone else in this thread is using it in this sense. I’m not sure what you mean by the word, but it’s clearly not the same thing everyone else in this thread means.

Good catch. Which is equivalent to your comment about gradient flow falling apart if it encounters a point of zero gradient, and (I believe) the wiki’s diagram of a circular gradient flow with an undefined center.