 Now, what we've looked at thus far with the continuity equation has been a derivation in Cartesian coordinates, but the continuity equation, just like any equations, you can use other coordinate systems. And so what we're going to do here is we're going to take a look at using the cylindrical coordinate system. And there's also a spherical, but we won't be looking at that. But you can look at spherical. Spherical would be a problem that you might have with a spherical geometry, but the continuity equation can be rewritten for other coordinate systems. And so what we're going to do, we're going to take a look at the cylindrical coordinate system. And I'll go to a new slide just because we might need a little bit more space. So cylindrical coordinates, what we'll begin by doing is drawing out some sort of baseline. And then I will draw the z-axis perpendicular to that. And then let's imagine we have some point up here. And that point would be at location r. So we'd have a vector r going to that point. And the angle that that point makes with respect to our baseline is theta. So it makes an angle of theta with respect to the baseline. And this particular point will give it coordinate location r, theta, and then z or z, denoting where it is along what we will call our cylindrical axis. So if we have that point, we can have velocity at that point. And with cylindrical coordinates, we show velocity in a slightly different manner. What we'll have is we will have one velocity component continuing on in the radial direction. So that would be vr. We will have velocity in a tangential direction. And that would be v theta. And then finally, we can have velocity going along our cylindrical axis. And in this case, that would then be vz. So those would be the three components of velocity that we would have at that particular point. The other thing we can have is quite often, we will write out differential elements, just like we did when we looked at deriving the continuity equation in Cartesian coordinates. So what I'll do is I'll begin by writing out a differential element. So if this is our differential element, what we can say is that this length here would be dz because it's in the cylindrical axis direction. This here on this side is dr because we have a coordinate r. And we have an angle here, d theta. So we have dr dz. And then the size of the inner part there would be rd theta. So I'll put that in red. So we would have rd theta would be that size there. And this is a typical infinitesimal element. Now, when we look at this, there are a few things that we can write out, transformations between Cartesian and our cylindrical coordinate system. Firstly, we can write r and express that in Cartesian. We can also express theta, the angle, as the arc tan. And finally, z maps directly to z. And so we don't have to do any transformation there. And in our relationship here, the way that we have this vr is the radial velocity. V theta is the azimuthal or circumferential because it is going around the circumference of a circle that you could draw at that point. And finally, vz is quite simply just the axial velocity. So those are the velocity components, as well as the coordinate transformation. With this, what we can do, we can rewrite our continuity equation. And in order to do that, what you need to know is the divergence operator in cylindrical coordinates. Sometimes I'll also call polar coordinates. So if we look at the divergence operator of vector a, which is the generic vector that we've been working with, and now I bring in time because it can be in time coordinates, but I'll say polar coordinates or cylindrical. Looking up in any math book, you'll find the divergence operator to be the following. So that's the divergence operator. And that is what we would need to put into our continuity equation. And with that, we can then rewrite continuity in cylindrical or polar coordinates. And it would look like this. So that is conservation of mass, cylindrical coordinates. And that is then the equation that you would apply if you have a problem involving cylindrical coordinates. For the most part, it's, you can see this term is the same. That is the same. But this and this are different. And if we were to look at spherical, it would be even a little bit more different than that. All three of the terms in the derivative there would be different for all three directions. So that is polar coordinates continuity. What we'll do in the next segment is we're going to apply this to a problem.