Fluid flow and vector fields

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 the 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 a little bit more here and if you imagine doing this it 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 for example this this particular one as you as you play the animation as you let the particles move along the vectors there's no change in the density and no point to 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 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 that pushes outward from everywhere and it's 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 be rotating around some points in this case counterclockwise but it's rotating clockwise around other still does that have any kind of mathematical significance does the fact that there seemed 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 a little warm-up to that as we're just visualizing multivariable functions