Divergence notation

- [Voiceover] So I've said that if you have a vector field, a two-dimensional vector field with component functions P and Q, that the divergence of this guy, the divergence of v, which is a scalar-valued function of x and y, is by definition, the partial derivative of P, with respect to x, plus the partial derivative of Q, with respect to y. And there's actually another notation for divergence that's kind of helpful for remembering the formula. And what it is, is you take this nabla symbol, that upside down triangle that we also use for the gradient, and imagine taking the dot product between that and your vector-valued function. And as we did with the gradient, the loose mnemonic you have for this upside down triangle, as you think of it as a vector, full of partial differential operators, and that sounds fancy but all it means is you kind of take this partial, partial x, a thing that wants to take in a function and take its partial derivative, and that's its first component, and the second component is this partial, partial y, a thing that wants to take in a function and take its partial derivative with respect to y. And, you know, loosely this isn't really a vector, these aren't numbers, or functions, or things like that, but it's something you can write down and it'll be kind of helpful symbolically. And you imagine taking the dot product with that, and, you know, v who has components, these scalar-valued functions, P of xy, and Q of xy. And when you imagine doing this dot product, and you're kind of lining up terms and the first one multiplied by the second, right, quote unquote multiplied, because, in this case, when I say this first component multiplied by p, I really mean you're taking that partial derivative operator partial, partial x, and evaluating it at p. That's kind of what multiplication looks like in this case. So, you take that, and as per the dot product you then add, what happens if you take this partial operator, this partial, partial y, and quote unquote, multiply it with q. Which, in the case of an operator, means you kind of give it the function q and it's gonna take its partial derivative. So, we see we get the same thing over here, it's the same formula that we have, and it's just kind of a nice, little, you can think of it as a mnemonic device for remembering what the divergence is. But another nice thing, this can apply to higher-dimensional functions, as well. Right? If we have something that, let's see, something that's a vector-valued function, and it's gonna be a three-dimensional vector field. So, it's got x, y, and z as its inputs, and its output then also has to have three dimensions. So, it might be like, P, Q, and R, and all of these are functions of x and y. So, that's P of x and y, Q, oh no, x, y, and z, right? So, P of x, y, kind of got in the habit of two dimensions there, P of x, y and z, Q of x, y, and z, and then R of x, y, and z. And I haven't talked about three-dimensional divergence. But if you think of this and then you imagine doing your nabla, dotted with the vector-valued function, it can still make sense. And in this case, that nabla you're thinking of as having three different components. It's gonna be, on the one hand this partial, partial x, I should say partial x there, partial x, now the second component is partial, partial y, and the last component is partial, partial z. And the ordering of these, of the variables, here, x, y and z is just whatever I have here. So, even if they didn't have the names x, y, z, you kinda out them in the same order that they show up in your function. And when you imagine taking the dot product between this, and your P as a function, Q as a function, and R as a function, vector-valued output, what you get, and I'll write it over here, you take that partial, partial x and kind of multiply it, with P, which means you're really evaluating at P. So, partial x here. Then you add partial, partial y. And you're evaluating at Q, because you're kind of imagining multiplying these second components. And you'll add what happens when you multiply by these third components, or that's partial, partial z, by that last component. And, you know, since I haven't talked about three-dimensional vector fields, with three-dimensional divergence, this last term, maybe it's not given that you'd have as strong an intuition for why this shows up in divergence as the other two, but it's actually quite similar, you're thinking about changes to the z component of a vector as the value z, of the input, as you're kind of moving up and down and that direction changes. But this pattern will go for even higher dimensions that we can't visualize, four, five, 100, whatever you want. And that's what makes this notation here quite nice, is that it encapsulates that and gives a really compact way of describing this formula that, it has a simple pattern to it, but would otherwise kind of get out of hand to write. See you next video.