Transformations, part 1 | Multivariable calculus | Khan Academy

So I have talked a lot about different ways that you can visualize multi-variable functions. Functions that will have some kind of multi-dimensional input or output. These include three-dimensional graphs, which are very common, contour maps, vector fields, parametric functions.

But here, I want to talk about one of my all-time favorite ways to think about functions, which is as a transformation. So anytime you have some sort of function, if you're thinking very abstractly, I like to think that there's some sort of input space, and I'll draw it as a blob, even though, you know, that could be the real number line, so it should be a line. Or it could be three-dimensional space.

And then there's some kind of output space, and again, I just very vaguely think about it as this blob, but that could be, again, the real number line, the xy-plane, three-dimensional space. The function is just some way of taking inputs to outputs. Every time that we're trying to visualize something, like with a graph or a contour map, you're just trying to associate input-output pairs. You know, if f inputs, you know, three gets mapped to the vector (1, 2), it's a question of how do you associate the number 3 with that vector (1, 2).

And the thought behind transformations is that we're just going to watch the actual points of the input space move to the output space. I'll start with a simple example that's just a one-dimensional function. It'll have a single variable input, and it'll have a single variable output. So let's consider the function f(x) is equal to x squared minus 3.

And of course, the way we're used to visualizing something like this, it'll be as a graph. You might kind of be thinking of something roughly parabolic that's squished down by three. But here, I don't want to think in terms of graphs; I just want to say how do the inputs move to those outputs.

So as an example, if you go to 0, when you plug in 0, you're going to get negative 3. You know, 0 squared minus 3 is equal to negative 3. So somehow we want to watch zero move to negative three. And then similarly, let's say you plug in one, and you'd get one squared minus three is negative two. So somehow we want to watch one move to negative two.

And just to list another example here, let's say you were plugging in 3 itself, so 3 squared minus 3 is 9 minus 3 is 6. So somehow, in this transformation, we want to watch 3 move to the number six. And with a little animation, we can watch this happen. We can actually watch what it looks like for all these numbers to move to their corresponding outputs.

So here we go; each number will move over and land on its output. I'll clear up the board here, so I kept track of what the original input numbers are by just kind of writing them on top here, and that was a way of just watching how it moves. And I'll play it again here; let's just watch where each number from the input space moves over to the output.

And with single variable functions, this is a little bit nice because it gives this sense of inputs moving to outputs. But where it gets fun is in the context of multi-variable functions. So now, let me consider a function that has a one-dimensional input and a two-dimensional output. And specifically, it'll be f(x) is equal to cosine of x and the y component will be x times sine of x.

So just to think about a couple examples, if you plug in something like zero and think about where zero ought to go, you would have f(0) is equal to cosine of 0, which is 1, and then 0 times anything is 0. So somehow we’re going to watch 0 move over to the point (1, 0), right? So this is where we expect 0 to land.

And let's think about like pi. So f(pi), and then cosine of pi is negative one. This is going to be pi multiplied by, and sine of pi is zero, so that'll again be zero. So, you know, this little guy is where zero lands, and we expect that this is going to be where the value pi lands.

And if we watch this take place and we actually watch each element of the input space move over to the output space, we get something like this. And again, this is just kind of a nice way to think about what's actually going on. You might ask questions about whether the space ends up getting stretched or squished.

And notice that this is also what a parametric plot of this function would look like if you interpret it as a parametric function. This is what you get in the end. But whereas in parametric plots you lose input information, here you can kind of see where things move as you go from one to the other.

And in the next video, I'm going to talk about how you can interpret functions with a two-dimensional input and a two-dimensional output as a transformation.