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Can someone help me understand how the laplacian of a function at a point relates to the average of the values of the points that surround it? (sh.itjust.works)

submitted 6 months ago* (last edited 6 months ago) by [email protected] to c/[email protected]

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I was watching this video, which was describing the intution behind the second derivative. I understand how the 1-dimmensional result was found, but I am quite at a loss for how to arrive at the n-dimmensional result, where the second derivative is the laplacian (the video provides the 3-dimmensional result, i.e. $f(x,y,z)$, at 00:08:08). The specific part that I'm having trouble with is finding the average of the multivariable function so that it fits the equation stated in the video, which, even more confusingly, has a single variable in it.

Update (2024-05-08T06:31Z): I found this document, which provides a lot more detail.

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[–] [email protected] 1 points 6 months ago (1 children)

The parametrization of x_0 representing a sphere on the curve seems needlessly abstract for the topic of the video. Even though it looks analogous to the one dimensional basis, it’s confusing you.

Hm, well, what confuses me about that is that the variables seem to be reused without stating that they are switching their meaning. I presume that $f(x)$ is a stand in for $f(x,y,z)$, and that $x_0$ is referring to the center point of the sphere. But, yeah, it is a strange jump to go from the basic curve to the abstract sphere idea.

Also the simile to curvature seems limiting

It certainly makes it difficult to use that sort of language when one is talking about >4-dimmensional shapes. For anything in 2-space, or 3-space, the terminology should function just fine.

I’m not aware of any widely used average of n-dimensional parametrization (other than the Laplacian and perhaps the normal), so the relation seems ill defined. You’ll have to check supplementary materials or alternate explanations.

Would you be aware of any rationale for why that might be? I was under the assumption that averages are defined regardless of the number of dimensions.

The summary formulation is practically useful for applications.

Which formulation are you referring to?

[–] [email protected] 1 points 6 months ago* (last edited 6 months ago) (1 children)

Would you be aware of any rationale for why that might be? I was under the assumption that averages are defined regardless of the number of dimensions.

I would rather invert the question, why would there be a broadly used definition unless it's useful?

Averages can also be defined in different ways, with medians, logarithmic means, squared means, Root Mean Square , and most of statistics, beyond the simple arithmetic one.

From an intuitive standpoint, how would you even go about to describe an n-dimensional curve as something useful in one dimension (scalar).

What meaningful, direction independent, quantity would you describe to differentiate between a cylinder and a sphere, or other shapes?

And I mean these questions not to be disparaging but as a guide, find a useful quantity and a meaningful formulation and you have created a mathematical tool. Get it to do cool and/or useful things, and people will start using it. As enough people adopt it, it becomes the widely used definition that we can point to the next time someone asks this question.

A simple way is to solve a practical problem with it, a more mathematical way is to express a neat relation, simplify a clumsy proof and/or solve one of the unsolved problems.

Normals are useful as they give the orientation of out (and are also useful in creating subspaces), so when working on finding the outside of surfaces, which we've spent many decades on, it came out as a commonly used tool. They're now also used in Machine learning theory to describe Solution Spaces to Parameter spaces, although I don't know enough to know if it's actually useful or just trendy.

Which formulation are you referring to?

The summary formulation: ∇² = f''(x) + f''(y) + f''(z).

[–] [email protected] 1 points 6 months ago

why would there be a broadly used definition unless it’s useful?

Knowledge for knowledge's sake? I find generalizations to be very satisfying.

Averages can also be defined in different ways, with medians, logarithmic means, squared means, Root Mean Square , and most of statistics, beyond the simple arithmetic one.

For the sake of this context, both me and the video are referring to the simple arithmetic mean.

From an intuitive standpoint, how would you even go about to describe an n-dimensional curve as something useful in one dimension (scalar).

Hrm, well, it would entirely depend on context.

What meaningful, direction independent, quantity would you describe to differentiate between a cylinder and a sphere, or other shapes?

I don't follow this point. Would it not be entirely context dependent?

And I mean these questions not to be disparaging but as a guide, find a useful quantity and a meaningful formulation and you have created a mathematical tool. Get it to do cool and/or useful things, and people will start using it. As enough people adopt it, it becomes the widely used definition that we can point to the next time someone asks this question.

Imo, this is approaching it from, say, a physicist's, or engineer's perspective, rather than from a mathematicians perspective. Math doesn't need to have direct applied use for it to be interesting.