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[SQUEAKING] [RUSTLING] [CLICKING] STEVEN G. JOHNSON: OK, so last time I talked about how in order to define a gradient, you need an inner product. So that way, if you have a scalar function of a vector, the gradient is defined-- basically the derivative has to be a linear function that takes a vector in and gives you a scalar out. So it turns out this has to be-- if you have a dot product, this has to be a dot product of something with the x. And we call the gradient. So the gradient is the thing with the same shape as x that we take the dot product with the x to get the f. So what I didn't mention is that, in fact, not only did we need a dot product to define a gradient, actually we swept something under the rug earlier. We actually need a norm in order to even define a derivative in the first place. All right. If you have a vector space, a norm is some measure of the length of the vector or a measure of distance. A norm takes in a vector v and gives you out a scalar...

[SQUEAKING] [RUSTLING] [CLICKING] STEVEN G. JOHNSON: OK, so last time I talked about how in order to define a gradient, you need an inner product. So that way, if you have a scalar function of a vector, the gradient is defined-- basically the derivative has to be a linear function that takes a vector in and gives you a scalar out. So it turns out this has to be-- if you have a dot product, this has to be a dot product of something with the x. And we call the gradient. So the gradient is the thing with the same shape as x that we take the dot product with the x to get the f. So what I didn't mention is that, in fact, not only did we need a dot product to define a gradient, actually we swept something under the rug earlier. We actually need a norm in order to even define a derivative in the first place. All right. If you have a vector space, a norm is some measure of the length of the vector or a measure of distance. A norm takes in a vector v and gives you out a scalar...

[SQUEAKING] [RUSTLING] [CLICKING] STEVEN G. JOHNSON: So I want to revisit the things that Alan talked about, but just a little bit more slowly and a bit more-- just try and lay out the rules for you as clearly as I can. And what we're going to try and do is, again, just revisit the notion of a derivative to try and write it in a way that we can generalize to other kinds of objects. And so I'm going to start with 18.01 and then go to 18.02 and so forth. So as Alan said, the key notion of a derivative, just, I think, it's easy to get so good at taking derivatives, like knowing the rule for the derivative of sine or cosine or x squared. You're so good at doing them that you forget what they are, right? And so the very first thing you learned about a derivative is that it's the slope of the tangent. But what that really is is linearization. So you have some arbitrary maybe nonlinear function f of x. And you're at a point x. And near that point, you...