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Multivariable functions

An introduction to multivariable functions, and a welcome to the multivariable calculus content as a whole. Created by Grant Sanderson.

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Video transcript

- [Voiceover] Hello and welcome to multivariable calculus. So I think I should probably start off by addressing the elephant in the living room here. I am, sadly, not Sal, but I'm still gonna teach you some math. My name is Grant. I'm pretty much a math enthusiast. I enjoy making animations of things when applicable, and boy, is that applicable when it comes to multivariable calculus. So, the first thing we gotta get straight is what is this word multivariable that separates calculus, as we know it, from the new topic that you're about to study? Well, I could say it's all about multivariable functions, that doesn't really answer anything because what's a multivariable function? And basically, the kinds of functions that we're used to dealing with, in the old world, in the ordinary calculus world, will have a single input, some kind of number as their input, and then the output is just a single number. And you would call this a single variable function. Basically because that guy there is the single variable. So then a multivariable function is something that handles multiple variables. So, you know, it's common to write it as x, y, it doesn't really matter what letters to use, and it could be, you know, x, y, z, x one, x two, x three, a whole bunch of things, but just to get started, we often think just two variables and this will output something that depends on both of those. Commonly it will output just a number, so you might imagine a number that depends on x and y in some way, like, x squared plus y, but it could also output a vector, right? So you could also imagine something that's got multivariable input, f of x, y, and it outputs something that also has multiple variables, like, I mean I'm just making stuff up here, three x and, you know, two y. And, this isn't set in stone, but the convention is to usually think if there's multiple numbers that go into the output, think of it as a vector, if there's multiple numbers that go into the input, just kind of write them, write them more sideways like this, and think of them as a point in space. Because, I mean when you look at something like this, and you've got an x and you've got a y, you could think about those as two separate numbers. You know, here's your number line with the point x on it somewhere, maybe that's five, maybe that's three, it doesn't really matter. And then you've got another number line and it's y, and you could think of them as separate entities. But, it would probably be more accurate to call it multidimensional calculus, because, really, instead of thinking of, you know, x and y as separate entities, whenever you see two things like that you're gonna be thinking about the x y plane. And thinking about just a single point. And you'd think of this as a function that takes a point to a number, or a point to a vector. And a lot of people, when they start teaching multivariable calculus, they just jump into the calculus, and there's lots of fun things, partial derivatives, gradients, good stuff that you'll learn. But I think first of all, I want to spend a couple videos just talking about the different ways we visualize the different types of multivariable functions. So, as a sneak peak, I'm just gonna go through a couple of them really quickly right now, just so you kind of whet your appetite and see what I'm getting at, but the next few videos are going to go through them in much, much more detail. So, first of all, graphs. When you have multivariable functions, graphs become three dimensional. But these only really apply to functions that have some kind of two-dimensional input, which you might think about as living on this x y plane, and a single number as their output and the height of the graph is gonna correspond with that output. Like I said, you'll be able to learn much more about that in the dedicated video on it, but these functions also can be visualized just in two dimensions, flattening things out. Where we visualize the entire input space in associated color, with each point. So this is the kind of thing where you, you know, you have some function that's got a two-dimensional input, that would be f of x, y, and what we're looking at is the x y plane, all of the input space, and this output's just some number, you know, maybe it's like x squared, this particular one is an x squared, but, you know that, and maybe some complicated thing, and the color tells you roughly the size of that output, and the lines here, called contour lines, tell you which inputs all share a constant output value. And again, I'll go into much more detail there. These are really nice, much more convenient than three-dimensional graphs, to just sketch out. Moving right along, I'm also gonna talk about surfaces in three-dimensional space. They look like graphs, but they actually deal with a much different animal, that you could think of it as mapping two dimensions, and I like to sort of spoosh it about. And we've got kind of a two-dimensional input, that somehow moves into three dimensions, and you're just looking at what the output of that looks like, not really caring about how it gets there. These are called parametric surfaces. Another fun one is a vector field, where every input point is associated with some kind of vector, which is the output of the function there. So this would be a function with a two-dimensional input and a two-dimensional output 'cause each of these are two-dimensional vectors. And the fun part with these guys is that you can just kind of, imagine a fluid flowing, so here's a bunch of droplets, like water, and they kind of flow along that. And that actually turns out to give insight about the underlying function. It's one of those beautiful aspects of multivariable calc. And we'll get lots of exposure to that. Again, I'm just sort of zipping through to whet your appetite. Don't worry if this doesn't make sense immediately. And one of my all-time favorite ways to think about multivariable functions is to just take the input space, in this case, this is gonna be a function that inputs points in two-dimensional space, and watch them move to their output, so, this is gonna be a function that also outputs in two dimensions. And I'm just gonna watch every single point move over to where it's supposed to go. These can be kind of complicated to look at, or to think about at first, but as you gain a little bit of thought and exposure to them, they're actually very nice, and it provides a beautiful connection with linear algebra. A lot of you out there, if you're studying multivariable calculus, you either are about to study linear algebra, or you just have, or maybe you're doing it concurrently, but understanding functions as transformations is gonna be a great way to connect those two. So with that, I'll stop jabbering through these topics really quickly and in the next few videos I'll actually go through them in detail and hopefully you can get a good feel for what multivariable functions can actually feel like.