Divergence formula, part 1

Hello everyone. So, now that we have an intuition for what divergence is trying to represent, let's start actually drilling in on a formula. The first thing I want to do is just limit our perspective to functions that only have an x component, or rather where the y component of the output is just zero.

So, this is some kind of vector field, and if there's only an x component, what this means is going to look like is all of the vectors only go left or right, and there's kind of no up or down involved in any of them.

So, in this case, let's start thinking about what positive divergence of your vector fields might look like near some point x, y. If you have your point, you know, this is the point x, y, somewhere sitting off in space.

Two cases where the divergence of this might look positive are: one, where nothing happens at the point, right? So, in this case, p would be equal to zero at our point. But then, to the left of it, things are moving to the left, meaning p, the, you know, the x component of our vector-valued function is negative, right? That's why the x component of this vector is negative.

But then, to the right, vectors would be moving off to the right, so over here, p would be positive. So, this would be an example of kind of a positive divergent circumstance where only the x component is responsible.

And what you'll notice here, this would be p starts negative, goes zero, then becomes positive. So, as you're changing in the x direction, p should be increasing. So, a positive divergence here seems to correspond to a positive partial derivative of p with respect to x.

And if that seems a little unfamiliar, if you're not sure how to think about, you know, partial derivatives of a component of a vector field, um, I have a video on that, and you can kind of take a look and refresh yourself on how you might think about this partial derivative of p with respect to x.

And once you do, hopefully, it makes sense why this specific positive divergence example corresponds with a positive partial derivative of p. But remember, this isn't the only way that a positive divergence might look. You could have another circumstance where, let's say, your point x, y actually has a vector attached to it.

So, this here, again, represents our point x, y, and in this specific example, this would be kind of p is positive. p of x, y is positive at your point there. But another way that positive divergence might look is that you have things coming in towards that point and things going away, but the things going away are bigger than the ones coming in.

But again, this kind of exhibits the idea of p increasing in value. You know, p starts off small; it's a positive but small component, and then it gets bigger, and then it gets even bigger.

So, once again, we have this idea of positive partial derivative of p with respect to x because changes in x, as you increase x, it causes an increase in p. This seems to correspond to positive divergence.

And you can even look at it if you go the other way, where you have a little bit of negative component to p here. So, p is a little bit negative, but to the left of your point, it's really negative, and then to the right, it's not nearly as negative.

In this case, it's kind of like as you're moving to the right, as x is increasing, you start off very negative and then only kind of negative, and then barely negative.

Once again, that corresponds to an increase in the value of p as x increases. So, what you'd expect is that a partial derivative of p, that x component of the output with respect to x, is going to be somewhere involved in the formula for the divergence of our vector field at a point x, y.

And in the next video, I'm going to go a similar line of reasoning to see what should go on with that y component.