One of our learning outcomes focuses on understanding flow of fluid across boundaries so that we can keep track of conservation of mass, conservation of momentum and conservation of energy. To understand what's going on, we need to look at a boundary in space. So let's have this line represent a boundary and we're interested in fluid that's traveling from A to B across this boundary and what it's taking with it, how much mass, momentum and energy. A flux is defined as a flow per unit area across a boundary. So that might be the mass per square meter of this little chunk of surface that's flowing through here. So it's a flow per unit area so it might be volume passing through a given surface area, mass passing through a given surface area or momentum passing through that given little element of surface area. If we zoom in on this little region of the surface, we might imagine that it's a straight line, as long as we're looking at a very small region, a straight line like this. And we can define a normal vector, outwards from the surface like this. So that's a little vector that's at 90 degrees locally to the surface. In here, the absolute flow velocity vector might not be perpendicular to the surface, it might be at some oblique angle to the surface. But we're interested if we want to know what's actually crossing the boundary, we're interested in the perpendicular velocity going across the boundary. So the local volume flux will be that perpendicular velocity. That's going to be meters per second. And if we had a one meter squared area here, then a velocity of one meter per second would take a couple of meters per second. We take one cubic meter of material per square meter of surface area across this boundary. So that would give us the flux. It's just the dot product of the velocity vector and the normal vector. And in this instance, because we've defined the normal vector pointing outwards from the surface, then it's negative V dot N gives us that perpendicular velocity that we're interested in. And that will be the volume flux through that surface. So the volume flow through a small area of surface, this little region here, will be the volume flux times the magnitude of that tiny area, DA. And we would need to integrate that if we wanted the total volume flow across that surface. And that dot product is just that perpendicular velocity. What we're really interested in is how much velocity do we have that's going directly across the surface, because that's what carries stuff from one side to the other. The volume flow through this whole surface will get by taking this little volume flow through a small surface and integrating it over the whole surface s. So we've still got the negative sign, we've got the V dot N, we've got DA, and we've got to integrate that over all of the little elements of area to get the volume flow. Now that may be geometrically quite complicated, but it's conceptually quite simple. So don't get hung up on the fact that we've got vector dot products and complexity here, just concentrate on the details. That's because we're going to move towards flat surfaces. So it's just a straight line. We're going to line up the surfaces that we're interested in parallel to the flow direction. We're going to consider an average velocity. And what is that average velocity? It's the result of integrating these velocity components over that whole area of interest. And we're going to wind up just saying that the flow volume, the volume flow rate, is the average velocity V integrated over this whole surface area times the total area of the surface. So that's going to get a little simpler for us as we go forward. But it all goes back to integrating over the entire surface area. So, we're going to go back to the surface area. And we're going to go back to the surface area. And we're going to go back to the surface area.