 The last thing that we're going to look at in this review of vector operators for the study of fluid mechanics is going to be relationships between the line, surface, and volume integrals. We use this sometimes depending upon the nature of the data that you might have, but these can be very useful tools either for analyzing experimental data, numerical data, or even doing analysis. So we're going to begin with one that links a line integral to a surface integral, and this is referred to as being Stokes's theorem. So let's imagine I'll draw out our little picture here. We have some surface, and if you recall, we were doing a surface integral, or a line integral, I should say, of that little surface. We had a differential element, d little s, and there was some vector field that we were looking at. We'll say that is vector a, and given that this is a surface, we have a curve going around it, and so that curve was c. With surface s, we also have defined some differential element, and there is a unit normal vector, and then the area, the scalar value of the area, is differential element d capital s. So through Stokes's theorem, what it enables us to do is relate a line integral around that surface. So we might be evaluating a dot ds, and that's a little s in this, and that is equivalent. We can relate that to being an area integral, and the relationship would be if you take the cross product of your vector field a that you're performing the line integral on, and then you dot that into the area vector ds, and so when we look at this, the dot product on the left hand side of this equation is going to result in a scalar term. Dot product always gives us a scalar, and on the right hand side, what we're going to get here is going to be a vector, and that is then dot product into a vector, and we know that the dot product in the dot product is going to result in a scalar. So what we get out of this is a scalar on the left and a scalar on the right, and then this is sometimes used for evaluating the circulation within a flow field, in that case we replace a with the velocity vector v, but that is Stokes's theorem, and sometimes you will see that in analysis. The next one that we're going to take a look at is a relationship between a surface integral and a volume integral. So these are related, the surface and the volume integrals of our vector field a through the divergence theorem, which is sometimes also called Gauss's theorem, and so let me draw out a schematic again of what we're looking at. And so in this case we have a closed surface encapsulating some volume v, and so that's closed surface capital S, and then we're going to have some vector and unit vector multiplied by the scalar value of our differential element ds, but through Gauss's theorem or the divergence theorem what we can write on the left hand side, we're going to have a closed integral over the surface, and in this case we're going to do the dot product, and that is equivalent on the right hand side to a volume integral, and the volume integral is going to be del dot a multiplied then by the differential volume element that we might be looking at within that volume, so in terms of what we're dealing with here we know that if we have a dot and a dot that's going to give us a scalar or a vector dotted by another vector, and then here we have grad dot a, so that is a scalar as well, and so the consequence of that is that we get a scalar on the left and a scalar on the right, and the final thing that we're going to take a look at is referred to as being the gradient theorem, and that relates a surface integral to a volume integral, and that's if we're dealing with a scalar field to begin with instead of a vector field, and so in this case if we're dealing with a scalar field again on the left hand side we have an area integral, but it's an integral with our scalar over the differential area element ds, and then on the right hand side what we end up with is a volume integral, and we deal with the gradient operator of that scalar field multiplied by the volume, and so looking at this on the left hand side we have a scalar multiplied by a differential element which is a vector, so we end up here with a vector, and then on the right hand side we had grad p, and then multiplied over the scalar so that as well will be a vector, so we have vector vector which is consistent, so those are relationships, Stokes's theorem, Gauss's theorem, and the gradient theorem, and you use them in different parts in fluid mechanics, if you see them that's where they're coming from, and sometimes depending upon the data that you might have you might use them in order to do your analysis, so that concludes the vector relationships in fluid mechanics, now what we'll continue doing is going on and looking at fluid mechanics from a differential analysis perspective.