 So in fluid mechanics we have line integrals, surface integrals, and volume integrals. We're now going to take a look at surface integrals. So imagine we have some arbitrary surface, and we'll call that capital S for the surface. And it is going to be bounded, this particular surface, let's assume that it's bounded by some curve C. So that would be the case if it's a two-dimensional, or it could be more than two-dimensional, but it's a, I guess, planar but curved surface. So it's not around a volume, I guess is what I'm trying to say. And we're going to have some differential element on here. We will call that dS, and it is going to have a unit vector normal to the surface d capital S, and we'll define d capital S as being that unit normal multiplied by the scalar magnitude of that little differential element of area. So in fluid mechanics we can have a number of different surface integrals. We can be doing surface integrals with scalars or with vectors. So we'll take a look at those here. And whenever you're doing a surface integral, quite often it'll be denoted with a two integral signs to denote the fact that you're integrating over a surface. And let's assume that we're doing a surface integral with a scalar to begin with. So when you do this, you take a scalar over the surface, and in this case it's an open surface, it will result in a vector. Now if what we're integrating is not a scalar, but it's a vector, there are two different ways we can do this, one with the dot product and the other with the cross, so let's take a look at both of those. So if our mathematical operation that we're looking at involves the dot product of a vector integrated over a surface, it will result in a scalar. Because we call the dot product always results in a scalar. If the mathematical operator that we're dealing with involves the cross product, so in that case we would be doing a surface integral of A cross ds, where ds is the area vector. So if we're dealing with a vector and we have the cross product as being our operator, the result will be a vector of our surface integral. We saw when we had the dot product, the result would be a scalar. And then if we have a scalar field and we're integrating over the area, the result of that would be a vector, because we have a scalar multiplied by a vector and it remains a vector. So those are surface integrals over open surfaces. Now it turns out that we have a closed surface, so what would be a closed surface? Maybe you're integrating among or across all of the surfaces of a cube, for example. And so if you wanted to integrate over all of those surfaces, that would be the case where we have what we call a closed surface integral. And so let's take a look at that. So if you're integrating around a volume, the only thing that would change is the way that we do the nomenclature. And quite often you'll see a closed double integral symbol like that. So that would be if you were integrating a scalar field, if you were integrating a vector field using the dot product operator. And then finally, if you were integrating a vector field but using the cross product operator. So that would be the way that you would denote the fact that you're integrating over a surface that happens to be a closed surface. And so that could be a sphere, a cylinder, a cube, anything like that. In the first case, that would be where you have an open surface where it doesn't close upon itself. So those are surface integrals. We use those periodically within fluid mechanics. And it is the middle of the looking at line integral along a curve, surface integral and volume. So the next segment, we'll take a look at volume integrals.