Transformations, part 3 | Multivariable calculus | Khan Academy

So I want to give you guys just one more example of a transformation before we move on to the actual calculus of multivariable calculus. In the video on parametric surfaces, I gave you guys this function here. It's a very complicated looking function; it's got a two-dimensional input and a three-dimensional output. I talked about how you can think about it as drawing a surface in three-dimensional space, and that one came out to be the surface of a donut, which we also call a Taurus.

So what I want to do here is talk about how you might think of this as a transformation. And first, let me just get straight what the input space here is. So the input space, you could think about it as the entire TS plane, right? We might draw this as the entire T-axis and the S-axis, and just everything here, and see where it maps. But you can actually go to just a small subset of that. So if you limit yourself to T going between zero and, let's say, 2 pi, and then similarly with S going from zero up to 2 pi, and you imagine what, you know, that would be sort of a square region, just limiting yourself to that, you're actually going to get all of the points that you need to draw the Taurus.

And the basic reason for that is that as T ranges from 0 to 2 pi, cosine of T goes over its full range before it starts becoming periodic. Um, S of T does the same, and same deal with S. If you let S range from 0 to 2 pi, that covers a full period of cosine, a full period of S, so you'll get no new information by going elsewhere.

So what we can do is think about this portion of the TS plane kind of as living inside three-dimensional space. This is sort of cheating, but it's a little bit easier to do this than to imagine, you know, moving from some separate area into the space. At the very least, for the animation efforts, it's easier to just start it off in 3D. Um, so what I'm thinking about here, this square is representing that TS plane, and for this function, which is taking all of the points in this square as its input and outputs a point in three-dimensional space, you can think about how those points move to their corresponding output points.

Okay, so I'll show that again. We start off with our TS plane here, and then whatever your input point is, if you were to follow it, and you were to follow it through this whole transformation, the place where it lands would be the corresponding output of this function. And one thing I should mention is all of the interpolating values, as you—in between these—don't really matter. A function is really a very static thing; there's just an input and there's an output.

And if I'm thinking in terms of a transformation, actually moving it, there's a little bit of, uh, a little bit of magic sauce that has to go into making an animation do this. And in this case, I kind of put it into two different phases to sort of roll up one side and roll up the other. It doesn't really matter, but the general idea of starting with a square and somehow warping that—however you do choose to warp it—is actually a pretty powerful thought.

And as we get into multivariable calculus and you start thinking a little bit more deeply about surfaces, I think it really helps if you, you know, you think about what a slight little movement over here on your input space would look like. What happens to that tiny little movement or that tiny little traversal? What it looks like if you did that same movement somewhere on the output space? Um, and you'll get lots of chances to wrap your mind about this and engage with the idea. But here, I just want to get your minds churning on this pretty neat way of viewing what functions are doing.