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[AUDIO LOGO] [MOUSE CLICK] STEVEN JOHNSON: OK, so let's get started. So as I said, I'm going to do second derivatives, which is just going to be the derivative of the derivative. When you have functions from matrices to matrices or something like that, you have to think a little bit carefully just to make sure we understand what kind of thing the second derivative is. So remember when we take basically the first derivative, what we have is this linear operator, f primed of x, that takes in a little change and gives us a df, which is the f of x plus dx minus f of x to first order, dropping higher-order terms. And so it's really natural to define the second derivative as the derivative of that. So what should f double prime be? f double primed of x should take in-- let me call it dx prime. I mean, that's not a derivative. It just means a different-- we put it in here in red. This is going to be a different small change. So prime here is not derivative, it...

[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...