2d curl intuition

hello everyone so I'm going to start talking about Curl Curl is one of those very cool vector calculus concepts and you'll be pretty happy that you've learned it once you have if for no other reason because it's kind of artistically pleasing and there's two different versions there's a two dimensional curl and a three dimensional curl and naturally enough I'll start talking about the two dimensional version and kind of build our way up to the 3d one and in this particular video I just want to lay down the intuition for what's what's visually going on and kernel has to do with the fluid flow interpretation of vector fields now this is something that I've talked about in other videos especially the ones on divergence if you watch that but just as a reminder you kind of imagine that each point in space is a particle like an air molecule or a water molecule and since what a vector field does is associate each point in space with some kind of vector and remember we don't we don't always draw every single vector we just draw a small sub sample but in principle every single point in space has a vector attached to it you can think of each particle each one of these water molecules or air molecules as moving over time in such a way that the the velocity vector of its movement at any given point in time is the vector that it's attached to so as it moves to a different location in space and that velocity vector changes it might be turning or it might be accelerating and that velocity might change and you end up getting kind of a trajectory for your point and since every single point is moving in this way you can start thinking about a flow kind of a global view of the vector field and for this particular example this particular vector field that I have pictured I'm going to go ahead and put a blue dot at various points in space and each one of these you can think of as representing a water molecule or something and I'm just going to let it play and at any given moment if you look at the movement of one of these blue dots it's moving along the vector that it's attached to at that point or if that vector is not pictured you know the vector that would be attached to it at that point and as we get kind of a feel for what's going on in this entire flow I want you to notice a couple particular regions first let's take a look at this region over here on the right kind of around here and just kind of concentrate on what's going on there and I'll go ahead and start playing the animation over here and what's most notable about this region is that there's counterclockwise rotation and this corresponds to an idea that the vector fee has a curl here and I'll go very specifically into what curl means but just right now you should have the idea that in a region where there's counterclockwise rotation we want to say the curl is positive whereas if you look at a region that also has rotation but clockwise going the other way we think of that as being negative curl here I'll start it over here and in contrast if you look at a place where there's no rotation where like at the center here you have some points coming in from the top right and from the bottom left and then going out from the other corners but there's no net rotation if you were to just put a you know if you were to put like a twig somewhere in this water it wouldn't really be rotating these are regions where you think of them as having zero curl so with that is a general idea you know clockwise rotation regions correspond to positive curl counterclockwise rotation regions correspond a negative curl and then no rotation corresponds to zero curl in the next video I'm going to start going through what this means in terms of the underlying function defining the vector field and how we can start looking at the partial differential information of that function to quantify this intuition of fluid rotation and what's neat is that it's not just about fluid rotation right if if you have vector fields in other context and you just imagine that they represent a fluid even though they don't this idea of rotation and curling actually has certain importance and in ways that you totally wouldn't expect the gradient turns out to relate to the curl even though you wouldn't necessarily think the gradient has something to do with fluid rotation in electromagnetism this idea of fluid rotation has a certain importance even though fluids aren't actually involved so it's more general than just the representation that we have here but it's a very strong visual to have in your mind as you as you study vector fields