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## Visualizing vector-valued functions

# Fluid flow and vector fields

## Video transcript

- [Voiceover] 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, and then these are going to start flowing in some way. How would you describe
this flow mathematically? So 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. 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, and 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 high upwards, whereas here, it's kind of
a smaller vector downwards, even though, I'll play the
animation a little bit more here, and 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, and 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 in. 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 punch
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. And over here, you see this vector field, and you might want to
understand what it's all about, and it's kind of helpful
to think of a fluid that pushes outward from everywhere, and is kind of decreasing in
density around the center, and 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 rotating around some points, in this case counter
clockwise, 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 there. What does that imply for the function that represents this whole vector field, and 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.