Explicit Laplacian formula

So let's say you have yourself some kind of multivariable function, and this time let's say it's got some very high dimensional input. So X1, X2, on and on and on, up to, you know, X sub n for some large number n.

Um, in the last couple videos, I told you about the Lassan operator, which is a way of taking in your scalar valued function f, and it gives you a new scalar valued function. It's kind of like a second derivative thing because it takes the divergence of the gradient of your function f. So the gradient of f gives you a vector field, and the divergence of that gives you a scalar field.

What I want to show you here is another formula that you might commonly see for this Leian. So first, let's kind of abstractly write out what the gradient of f will look like.

So we start by taking this Dell operator, which is going to be a vector full of partial differential operators: partial with respect to X1, partial with respect to X2, and kind of on and on and on up to partial with respect to whatever that last input variable is. You take that, that whole thing, and then you just kind of imagine multiplying it by your function.

Uh, so what you end up getting is all the different partial derivatives of f, right? It's partial of f with respect to the first variable, and then kind of on and on and on up until you get the partial derivative of f with respect to that last variable X sub n.

And the divergence of that, and just to save myself some writing, I'm going to say you take that n operator and then you imagine taking the dot product between that whole operator and this gradient vector that you have here.

What you end up getting is, well, you start by multiplying the first components, which involves taking the partial derivative with respect to X1, that first variable, of the partial derivative of f with respect to that same variable.

So it looks like the second partial derivative of f with respect to that first variable, so the second partial derivative of f with respect to X1, that first variable. And then you imagine kind of adding what the product of these next two items will be.

For very similar reasons, that's going to look like the second partial derivative of f with respect to that second variable, partial X2 squared. And you do this to all of them, and you're adding them all up until you find yourself, you know, doing it to the last one.

So you've got plus and then a whole bunch of things, and you'll be taking the second partial derivative of f with respect to that last variable, partial of X sub n. This is another format in which you might see the Lan, and oftentimes it's written, um, kind of compactly.

So people will say the Lan of your function f is equal to, and then using Sigma notation, you'd say the sum from I is equal to 1 up to, you know, 1, 2, 3, up to n. So the sum from that up to n of your second partial derivatives, partial squared of f with respect to that e variable.

So, you know, if you were thinking in terms of three variables, often X1, X2, X3, we instead write XYZ, but it's common to more generally just say X sub I. So this, this here is kind of the alternate formula that you might see for the Lan.

Personally, I always like to think about it as taking the divergence of the gradient of f because you're thinking about the gradient field, and the divergence of that kind of corresponds to maxima and minima of your original function.

Which is what I talked about in the initial intuition of the Lassan video. But this, this formula is probably a little more straightforward when it comes to actual computations.

And oh wait, sorry, I forgot I squared there, didn't I? So, uh, partial X squared. So this is the second derivative. Yeah, so summing, summing these, uh, second partial derivatives, and you can probably see this is kind of a more straightforward way to compute a given example that you might come across.

And it also makes clear how the Lan is kind of an extension of the idea of a second derivative. See you next video.