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[SQUEAKING] [RUSTLING] [CLICKING] STEVEN JOHNSON: So Alan mentioned fine differences as something that automatic differentiation is not doing. But I do want to talk about fine differences because they do come up a lot when you're talking about computing derivatives on a computer. And as he mentioned, we're going to spend a lot of time talking about automatic differentiation, which is this amazing technology where you can take a program that computes a function. And it will basically compute the derivative analytically in some sense for you. But how it does it takes a little bit of explanation. And this is great. If you're hand-computing derivatives, even though you might think the rules are simple, as you see from the homework, when you get to more complicated functions, especially involving vectors and matrices, it's quite error-prone. And it's a common source of bugs if you're doing numerical optimization route finding sensitivity analysis-- all ...

[SQUEAKING] [RUSTLING] [CLICKING] STEVEN JOHNSON: So I want to briefly start to talk about going beyond 18.02 derivatives. So we've already gone, I would say, beyond 18.02 in the sense that they never do the-- they have Jacobians maybe, but they never really do the chain rule. They never really write even this definition in-- where is it? Go back-- saying that the change in the function is the Jacobian times dX, which I view as almost the definition of the Jacobian. I think that they never really do that. They never really do, like, multi-dimensional Taylor expansion or something like that. But now what we want to do in this class is go really beyond 18.02 to have functions where the inputs and outputs are not just column vectors necessarily or scalars. But they could live in other vector spaces, in sort of more general vector spaces. So let me give an example. And you probably learned in 18.06 hopefully, remember, that a vector space is not just a column vector. It...