 So today we're going to talk a little bit about volume and the relationship of rate of change of volume, which is called flow. Volume is a measurement, is our measurement of space in three dimensions. And for solids, solid materials, the boundaries, for example I have a baseball here, the boundaries of a solid are well defined, that's part of what defines a solid is something that has a definite shape. And the motion of something that's solid, if we take something like this ball and we move it, the motion of that is usually considered called rigid body motion because it's assumed that the body maintains that shape. And often when we analyze something like that, we travel with the body. We move with the body, we consider thinking about things with the body. This is called, has a special term called Lagrangian motion or Lagrangian perspective where we think about that motion as moving with whatever we're analyzing. So now when we talk about fluids, fluids are a little bit different, fluids being both liquids and gases. In their case, they don't have a definite shape and what ends up defining our shape for a fluid is usually some sort of solid container. For example, if here I have a lake, I have sort of the bottom of the lake that's sort of containing the lake. Or we can put a conceptual boundary or define some sort of shape. For example, if I'm looking at the bottom of the lake, I might create a column of water. And I might sort of create an area that I'm conceptually thinking about that is the column of water. And then the third thing we can sort of think about is we can have just a general conception where maybe we have a box here and in that box or pool that's going to represent all of the lake water in North Carolina. And notice that doesn't necessarily have a shape but it sort of has a conceptual set of boundaries around it that captures everything that we're talking about. So if I let the shape sort of change but I move with the shape, then that's still sort of a Lagrangian perspective. For example, if there's a stream, if there's water flowing in a stream and I make some dye and sort of put a plug of dye in that water and then I watch and I travel with that dye as it moves to the stream and maybe the dye stretches out and gets thinner as it moves downstream. And if I kind of travel with that dye, that is still a Lagrangian perspective. However, there's another perspective that we often use with fluids. In this case, the other perspective is something called the Eulerian perspective. And the Eulerian perspective, what we're trying to consider is the idea of we use our boundaries but we let the fluid flow in and out of the boundaries. For example, with our lake here, if we have a stream that's running out and perhaps rain or runoff from the side of a mountain that's flowing into the stream, we might not keep track, we may not follow what happens to the water when it leaves and we may not know where it's coming from. If we just consider the water within the boundaries, that is our Eulerian perspective where we kind of encase ourselves. Similarly with this column of water, there may be water leaving, flowing through and at the same time, water flowing in. Or if we look at our conceptual idea here, the lake water in North Carolina, we can think about the things that might add to that lake water or subtract from that lake water. For example, precipitation might add to the overall lake water whereas we might have some sort of streams flowing off, removing the overall lake water, and we might have evaporation, removing the overall lake water. For each of these things, for each of these elements, inside we have some sort of volume, a volume V. There's a volume inside the lake, there's a volume here inside my column of water and there might be some sort of volume of water encased representing all the lake water in North Carolina.