Multivariable functions | Multivariable calculus | Khan Academy

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 S, but I'm still going to teach you some math. My name is Grant. Um, 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 got to 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? Uh, well, I could say it's all about multivariable functions. That doesn't really answer anything because what's a multivariable function?

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 it outputs just a single number. 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 like x, y. It doesn't really matter what letters to use. It could be, you know, x, y, z, x1, x2, x3, a whole bunch of things. Um, but just to get started, we often think just two variables, and this will output something that depends on both of those. Um, commonly, they output just a number.

So you might imagine a number that depends on x and y in some way, like x² + y. Uh, but it could also output a vector, right? So you could also imagine something that's got multivariable input, f(x, y), and it outputs something that also has multiple variables. Like, I mean, I'm just making stuff up here: 3x and, you know, 2y.

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 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 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 multi-dimensional calculus because really, instead of thinking of, you know, x and y as separate entities, whenever you see two things like that, you're going to be thinking about the xy-plane and thinking about just a single point.

You'd think of this as a function that takes a point to a number or a point to a vector. 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, um, good stuff that you'll learn.

But I think first of all, I want to spend a couple of videos just talking about the different ways we visualize the different types of multivariable functions. So as a sneak peek, I'm just going to go through a couple of them really quickly right now, just so you kind of wet 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 xy-plane, and a single number as their output. The height of the graph is going to correspond with that output.

Um, like I said, you'll be able to learn much more about that in the dedicated video on it. Uh, but these functions also can be visualized just in two dimensions, flattening things out where we visualize the entire input space and associate color with each point.

So this is the kind of thing where you know you'd have some function that's got a two-dimensional input. It would be f(x, y), and what we're looking at is the xy-plane, all of the input space. And this outputs just some number. You know, maybe it's like x². This particular one is an x², but you know that and maybe some complicated thing.

Um, 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.

Um, moving right along, I'm also going to 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 as mapping two dimensions. And I like to sort of spush 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.

Um, not really caring about how it gets there. Um, 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 then two-dimensional output because 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 calculus, and we'll get lots of exposure to that.

Again, I'm just sort of zipping through to wet your appetite. Don't worry if this doesn't make sense immediately. One of my all-time favorite ways to think about multivariable functions is to just take the input space. In this case, this is going to be a function that inputs points in two-dimensional space and watch them move to their output.

So this is going to be a function that also outputs in two dimensions, and I'm just going to watch every single point move over to where it's supposed to go. Um, 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. Uh, but understanding functions as transformations is going to be a great way to connect those two.

So with that, I'll stop jabbering through these topics really quickly. 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.