 So let's talk a little bit about stream flow, or actually flow through any sort of conduit or channel, and the relationship between flow and the speed of the fluid that is flowing through the channel. So let's consider the flow of water through a channel. Maybe this is a stream. Notice there's sort of a little cut in there. And we're representing the flow of water, again, the volumetric flow of water with the letter q. And I have three views here. You can sort of see the top view, the side view, and the front view sort of representing here all the different perspectives of our isometric view of this water. And we're only going to consider a chunk of this stream. We're going to slice into the stream and consider a chunk of it. It continues to flow down here, and it continues to flow in from up this level up top. But we're only going to consider some sort of chunk of that flow of water. And if I look at it here from the side, I'm going to take this side perspective and think about this chunk of water from the side. And if it's flowing, it means after some period delta t, if I wait for a little bit of time, the water that was in the chunk that we're thinking about is going to have moved downstream a little bit. So here I have the chunk, and then here I have the outline of where that volume of water was and where it is now. And notice in that time frame, it's going to have moved some small distance that will label delta x. And now the water may not all move at the same speed, but in general, the entire chunk of water will be moving on average a certain distance x. And that means that the speed that the water is moving, our velocity of the water is going to be equal to delta x over delta t. That's how fast all the water is moving downstream. So notice there is a change in the amount of water that's in our original boundaries in our original area. There are some that must have flowed out. Here's the water that's flowed out. And we're assuming that water will also flow in. But let's just consider the water as it flows out. If I look at this little wedge section here, notice I'm sort of cutting off a little section that's in front. Consider this as being a little section right here. I'm going to translate it down here. That distance that the water flowed, there's my delta x where the new water is sort of extending out from our little section there. And if we look at where the water is, we have the cross-section of the water, A. The cross-sectional slice where the water is basically where we're looking at the water. We're cutting across this cross-section. It has a cross-sectional area, A. Well, if I think about this as being a prism, like a cylinder or a rectangular prism, in this case it's sort of shaped like a triangle, I can use the relationship that the volume. And I'm going to use a little slash here on the bottom of my volume to help distinguish it from being the velocity v. And in fact, maybe I will try to add a little bit of cursive lettering to the velocity v to help distinguish those two. But if I have my volume v, and I can basically relate that by taking it, multiplying the area, this cross-sectional area here, times the thickness of this little wedge, delta x. Notice from our earlier relationships we know that our flow rate Q is the change in volume. And in fact, this volume I'm talking about is a change in volume. Notice it's the amount of water that's flowed out. Here's our little delta volume, the water that's flowed out from past our boundaries. So this change in volume is equal to the area times that change in x. I know that my flow rate is equal to the change in volume over the change in time. Well, if I plug that change of volume in here, that's equal to the cross-sectional area times that little position that it shifted divided by the change in time. And we've just defined that change in x, that shift in time as our velocity up here. So we can say a times small v, our velocity here. In other words, the flow rate is equal to the cross-sectional area times the velocity. Now that velocity is an average speed. It's an average speed of the water as it flows through. Let's check our unit, shall we? Let's just make sure that our units make sense. If you remember, our units for our flow rate are volume meters cubed as an example over time. Let's say it's in seconds, for example. Whereas our units for area are something like square meters, and our units for speed would be something like distance meters over time. Notice meters squared times meters is going to be the equivalent of meters cubed and seconds times seconds, well, there's seconds on the bottom in each case. So in both cases, we have meters cubed per second and meters cubed per second. So our units do indeed check out. This equation here, or this relationship here, is sometimes known as the continuity equation. And usually, we apply it in a situation where our flow in is equal to our flow out. Remember, that's our conditions for equilibrium in an Eulerian system, where the amount of flow in, here's our flow in, is equal to our flow out. If we do not want the overall volume to change, and something like a stream, generally, we'll consider equilibrium conditions. Well, if this is the case, qn equals q out, one of the relationships we can use by plugging in our new AV relationship is that our area in times the velocity in is equal to the area out times the velocity out.