 We're now going to start taking a look at the second main technique in the analysis of fluid mechanics, and that is the area of differential relations for a fluid particle. So this is the second main technique, small scale analysis, but what we've looked at thus far, we started by looking at applying the basic equations of physics, so conservation of mass, momentum, as well as energy to a finite control volume. So what we looked at thus far was this. And control volume analysis is sometimes also called integral analysis, and the reason for that is you're not able to determine the exact velocity profile. You're usually given a velocity profile. So it is integral analysis, and we're doing integration of a known velocity profile. And it's also kind of referred to as being large scale analysis. And the reason is it enables us to determine things such as forces, acting on objects and moments, things like that. What we're now going to move into is going to be giving us more detail about the fluid mechanic or about the flow field that we're examining. And what we're going to do here, we're going to start to apply the conservation laws to a little finite differential or infinitesimal control volume, and that'll be our starting point. So we're going to build from what we already know about control volume analysis. So we're going to apply the conservation laws to an infinitesimal or very, very small control volume, and this leads us into the area of differential analysis. And it's really more concerned with fluid properties at a point-by-point basis. And one comment about differential analysis of fluid mechanics is that the governing equations that we're going to derive been known for hundreds of years, Euler's equation, Navier-Stokes equations, and there've only been a limited number of analytical solutions that we've been able to come up with. But it's been in the last 50 years or so with the advent of the digital computer that it enables us to be able to do a lot more with the differential form of the equations. And so really in the last 50 years, the equations that we're going to look at have been kind of brought to life by the computer. So Navier-Stokes equation, Euler's equation, like I said, they've been known for a long, long time. But with the digital computer, what has happened is we have a bit of a revolution in the field of fluid mechanics. And what has resulted is the field called computational fluid dynamics. And this is CFD, sometimes the acronym. And we're not going to get into this in this course, but what you should know is the equations that we're going to derive become the basis for any kind of CFD approach. And one of the main obstacles, obviously, is turbulence, which we will just touch on briefly in the course. Even CFD is having a hard time tackling turbulence, and anyways, that's another story. I won't get into that now. But for laminar flow, certainly CFD can tackle any kind of flow field that we're interested in. And the equations that we look at will be the basis of that. So what we're going to do, we're going to begin with continuity, and we will be applying the control volume analysis to that. But for an infinitesimal element, a tiny one, and then we'll work our way through towards momentum. And that will lead us to the Euler equation, and then finally Navier-Stokes.