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.
I’m not a fan of their approach to explaining it, but if you look at 08:50, you’ll see that the Laplacian² is just the sum of second derivatives in each of your dimensions: f’‘(x) + f’‘(y) + f’'(z).
I couldn’t stomach watching the whole video, so I don’t know how they define <f>, and the parametrisation of x_0 representing a sphere on curve seems to become needlessly abstract for the topic of the video.
What specifically are you referring to? Also, out of curiosity, why don’t you like their explanation?
Well, yes, that is how the laplace operator is defined. That doesn’t answer the overarching question, though. Also, just to be pedantic for a moment, saying “Laplacian²” is innacurate. I think you are referring to the nabla as a “laplacian”, but it’s not called that — the proper term is “nabla”, or “del”. The operator that is written as a nabla with the superscript of 2 is, in its entirety, referred to as the laplacian.
They don’t (for n-dimensions). That was, essentially, the main issue that I was having.
The parametrisation 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.
Also the simile to curvature seems limiting (as evidenced by the confusion at the abstraction to higher dimension), but maybe that’s just me struggling to visualise n-dimensional shapes.
You are correct.
I’m not aware of any widely used average of n-dimensional parametrisation (other than the Laplacian and perhaps the normal), so the relation seems ill defined. You’ll have to check supplementary materials or alternate explanations.
The summary formulation is practically useful for applications.
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.
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.
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.
Which formulation are you referring to?
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.
The summary formulation: ∇² = f’‘(x) + f’‘(y) + f’'(z).
Knowledge for knowledge’s sake? I find generalizations to be very satisfying.
For the sake of this context, both me and the video are referring to the simple arithmetic mean.
Hrm, well, it would entirely depend on context.
I don’t follow this point. Would it not be entirely context dependent?
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.