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Current time:0:00Total duration:5:23

Local linearity for a multivariable function

Video transcript

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 X Y and it's also going to spit out a 2d vector and the specific one I have in mind this is just kind of arbitrary is X plus sine of Y and then because I'm a sucker for symmetry I'm going to make it Y plus sine of X now of course you know this could be any arbitrary function you don't need that kind of symmetry so 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 use this grid and here I want to show what this function looks like as a transformation of space as then I'm going to tell the computer 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 plus sine of Y y plus sine 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 there are 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 and 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 0 ok so if that's what I'm plugging in X is PI halves so at the top here X stays the same it's probably halves and then sine of Y would be sine of 0 so that X component is going to completely stay the same and then for the bottom Y well Y is also 0 plus sine of X sine of PI halves is 1 I'm just I'll go ahead and write sine of PI halves sine of PI halves but you can think of that as just being 1 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 is 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 for numbers to record where everything goes but this function has a nice property a property that we deal with all the time and multivariable calculus it's what we'd call locally linear locally linear and what that means is if I was to take our initial set up 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 gridlines so they're really very close gridlines 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 to follow the point edit 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 gridlines 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 gridlines right around it so they're very very densely packed and this is 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 gridlines and only the gridlines and where they move gives sort of a feel for what the function is 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 the neighborhood around it all of the points around it really really do look like a linear function and 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 naught and y naught this should correspond in some way to the linear transformation that it looks like around it there should be some kind of matrix some two-by-two 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