Fluid flow and vector fields | Multivariable calculus | Khan Academy

So in the last video, I talked about vector fields, and here I want to talk about a special circumstance where they come up. So imagine that we're sitting in the coordinate plane, and that I draw for you a whole bunch of little droplets, droplets of water.

Then these are going to start flowing in some way. How would you describe this flow mathematically? At every given point, the particles are moving in some different way. Over here, they're kind of moving down and to the left; here, they're moving kind of quickly up; and over here, they're moving more slowly down.

So what you might want to do is assign a vector to every single point in space. A common attribute of the way that fluids flow—this isn't necessarily obvious—but if you look at a given point in space, let's say like right here, every time that a particle passes through it, it's with roughly the same velocity.

So you might think over time that velocity would change, and sometimes it does. A lot of times there's some fluid flow where it depends on time, but for many cases, you can just say at this point in space, whatever particle is going through it, it'll have this velocity vector.

So over here, they might be pretty like high upwards, whereas here it's kind of a smaller vector downwards. Even though—and here I'll play the animation a little bit more—if you imagine doing this at all of the different points in space and assigning a vector to describe the motion of each fluid particle at each different point, what you end up getting is a vector field.

So this here is a little bit of a cleaner drawing than what I have, and as I mentioned in the last video, it's common for these vectors not to be drawn to scale but to all have the same length just to get a sense of direction. Here you can see each particle is flowing roughly along that vector.

So whatever one it's closest to, it's moving in that direction. And this is not just a really good way of understanding fluid flow, but it goes the other way around; it's a really good way of understanding vector fields themselves.

So sometimes you might just be given some new vector field, and to get a feel for what it's all about—how to interpret it, what special properties it might have—it's actually helpful, even if it's not meant to represent a fluid, to imagine that it does, and think of all the particles, and think of how they would move along it.

For example, this particular one, as you play the animation, as you let the particles move along the vectors, there's no change in the density. At no point do a bunch of particles go inward or a bunch of particles go outward; it stays kind of constant.

And that turns out to have a certain mathematical significance down the road. You'll see this later on as we study a certain concept called divergence. Over here, you see this vector field, and you might want to understand what it's all about. It's kind of helpful to think of a fluid that pushes outward from everywhere, and this kind of decreasing in density around the center.

That also has a certain mathematical significance, and it might also lead you to ask certain other questions. Like, if you look at the fluid flow that we started with in this video, you might ask a couple questions about it.

Like, it seems to be rotating around some points—in this case, counterclockwise—but it's rotating clockwise around others still. Does that have any kind of mathematical significance? Does the fact that there seem to be the same number of particles roughly in this area but they're slowly spilling out, what does that imply for the function that represents this whole vector field?

You'll see a lot of this later on, especially when I talk about divergence and curl. But here, I just wanted to give a little warmup to that as we're just visualizing multivariable functions.