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?
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?
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.
1I’m not a fan of their approach to explaining it
What specifically are you referring to? Also, out of curiosity, why don't you like their explanation?
if you look at 08:50, you’ll see that the Laplacian² is just the sum of second derivatives in each of your dimensions
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.
so I don’t know how they define <f>
They don't (for n-dimensions). That was, essentially, the main issue that I was having.
1What specifically are you referring to? Also, out of curiosity, why don't you like their explanation?
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.
The operator that is written as a nabla with the superscript of 2 is, in its entirety, referred to as the laplacian.
You are correct.
They don't (for n-dimensions). That was, essentially, the main issue that I was having.
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.
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