 So we've looked at the rotation in a fluid particle and it turns out that rotation is not often the way that we describe that type of motion within a fluid particle. What we often do is we refer to the vorticity, but vorticity is related to the rate of rotation that we computed. And the relationship is we use capital omega and that's a vector is defined as being equal to 2 times the rotation rate that we computed. And if you recall back we said that the rotation rate was one half of the curl of the velocity vector, so it's then equal just to del cross v. And that would give us the vorticity in the fluid. And what this is, it's a measure, so vorticity is a measure of the rotation of a fluid element as it moves through the flow. And vorticity is related to another variable or thing that we calculate within fluid mechanics. And that is the circulation. Circulation is used quite often in aerodynamics, but it is defined as being capital gamma. It is a scalar and it is evaluated as being a line integral of the velocity vector around some closed path. And so a circulation, a line integral of the tangential velocity about a closed curve in the flow that is fixed. So what we're going to do, let's compute the circulation in a fluid particle. And so what we're going to do, we're going to look at the same one that we looked at for fluid rotation. So there we have a flow field that we're going to look at. And what we want to do, we want to evaluate the circulation about that curve. And so in order to compute the circulation, looking back at the definition of circulation, so this is a differential element. It's going to be the integral of v.ds as we go around the curve. So I'm going to start section by section. We will label these. This will be the first, then we'll do that second. This will be the third, and then finally that the fourth. And that will take us all the way around the curve. So that is what the circulation is for this differential element. It's a small fluid element that we're looking at. And so I can rewrite that by combining some terms. So we get that for the circulation. And you'll notice that the term in brackets here, that looks a lot like our rotational rate that we computed earlier. So I'm going to make a substitution. And we get that as well for the circulation. And so with that, what we can do is we can make an inference because it's showing that a line integral is in a way related to our rotational rate multiplied by this here was an area component. So we have seen a thing when we looked at vectors and through vector review, there was an equation that looks somewhat like this. And that equation is Stokes theorem. And what it does is it relates this line integral to an area integral of the vorticity. So the velocity integral is equal to the vorticity and integrated over the area that we're looking at. So what that tells us is that the circulation around a closed contour is the sum of the vorticity enclosed within it. So those two are equivalent to one another. Places where you may apply that, you may have a velocity field that you've measured experimentally. Quite often nowadays, we have particle image velocimetry, which will give you the velocity field in a plane. And so you're going to get a bunch of velocity vectors in here that you're measuring at a given point in time. What you can do with those velocity vectors is you can convert them into your vorticity. I'll be two times the rotational rate. So I'll put it as the vorticity in the z direction. And that would then give you an inference as to the circulation in the flow. So if you have a big, large scale organized vortex, it would be one way of being able to quantify it. You would use PIV to get the velocity field, and then that would give you an inference as to the circulation around that vortex. Many different applications in that, but that would just be one that you might get out of experimental fluid mechanics. So that is vorticity and circulation, and they relate to the rotation within our fluid elements.