Local linearity for a multivariable function

So a lot of the concepts that you learn about in multivariable calculus are really all about taking ideas that you originally might have learned in linear algebra and then transferring those to apply to nonlinear problems.

So for example, I'm going to give you a function, some kind of function that takes in a 2D vector XY, and it's also going to spit out a 2D vector. The specific one I have in mind, this is, uh, just kind of arbitrary, is X Plus s of Y. And then because I'm a sucker for symmetry, I'm going to make it Y Plus s of X. Though of course you know this could be any arbitrary function; you don't need that kind of symmetry.

In the last video, I gave a little refresher on how to think about linear transformations and ideas from linear algebra and how you encode a linear transformation using a matrix and kind of visualize it. I used this grid, and here I want to show what this function looks like as a transformation of space.

As in, I'm going to tell the computer to take every single point on this blue grid here, and if that point is XY, I want you to move it over to the point X + s of Y, Y + S of X. And here's what that looks like.

All right, so things get really wavy, really curly. This is not at all a linear transformation, right? All of the lines don't remain lines; they're no longer nice grid lines that are parallel and evenly spaced. In some sense, there is much, much more information that goes into nonlinear functions than into linear functions.

Because this is rather complicated, I think it might be easier to see what's going on if we just focus on a single individual point. So let me look at a point like, let's say, Pi halves and zero.

Okay, so if that's what I'm plugging in, X is pi halves. So at the top here, X stays the same; it's Pi halves. And then s of Y would be S of zero, so that X component is going to completely stay the same. And then for the bottom Y, well, Y is also zero plus sin of X. S of Pi halves is one; I'm just, I'll go ahead and write s of Pi halves.

But you can think of that as just being one. So what that means on the transformation over here is if we look at the point that's at Pi halves, 0, and Pi halves is a little above 1.5, so that's going to be around here. We expect it to move to the point Pi halves, 1, so it should just move vertically by one unit.

And if you just focus on that one point during the transformation, notice that's exactly what happens; it just moves vertically one point. And of course, things are quite complicated because every point is doing that, right? The computer's taking in every point and moving it to where it should go.

So after having given the refresher on thinking about linear transformations and encoding them with matrices last time, something like this might feel completely intractable. You certainly have to store much more information than just four numbers to record where everything goes.

But this function has a nice property—a property that we deal with all the time in multivariable calculus. It's what we call locally linear. Locally linear. And what that means is if I was to take our initial setup and then zoom in on a given point, so I'm going to zoom in around this point on the left here, and this box kind of in the upper right just shows the zoomed-in version of that.

And first of all, I'm going to add some more grid lines, so they're really very close grid lines, right? We can see from the zoomed-out picture, but this just makes it so that when we're zoomed in, we can see a little bit more of what's going on.

And now when I play the animation, I'm going to have this yellow box that's doing the zooming follow the point at its center, right? So this box will be moving, and we're always just going to look at what it zoomed in on.

Okay, so it's going to be following what's going on around that point during the transformation, and we can see inside this zoomed version; it's still not linear, right? The lines get a little bit curved, but this looks a lot more like a linear function.

It looks a lot more like the grid lines that started off horizontal and vertical are remaining parallel and evenly spaced. And in fact, let's say I zoom in even further to an even smaller yellow box here, and again I'm going to add in some more grid lines right around it, so they're very, very densely packed.

And this is purely an artifact of visualizing things, right? I could choose to put points or lines or anything wherever I want, and I just think showing the grid lines and only the grid lines and where they move gives sort of a feel for what the function is doing.

So this time when I play it and that zooming in box kind of tracks the point that we're looking at as it goes to the neighborhood around it, all of the points around it really, really do look like a linear function. The more you zoom in, the more it looks precisely like a certain linear function.

Oh, I guess I should have written an R over here—locally linear. So this raises the question: if we're looking around some specific point, which I'll call X knot and Y knot, this should correspond in some way to the linear transformation that it looks like around it.

There should be some kind of matrix, some 2x2 matrix, that represents the linear transformation that this function, this much more complicated function, looks like around that point.

So this idea of zooming in is what we mean by local. And in the next video, I'm going to show you what this matrix looks like in terms of partial derivatives for our original function. See you then.