 We're now going to take a look at some of the integrals that we use in fluid mechanics and we'll begin with the line integral. And so what we'll do, we'll begin by defining some arbitrary vector field A and we'll also define a curve and the curve is where we're going to be doing the line integral on so we'll call this curve C and we'll say the curve is going in space from A to B and along that curve we will have some differential element ds and there will be a vector in the direction of ds. I'll put a unit vector in and over this differential element let's assume that the vector, our vector field A is this. So it has some value over that small differential element and let's write ds in terms of a vector so ds written in vector form would then be the scalar value multiplied by the unit vector that we set in the direction n. So let's assume that we're wanting to evaluate the line integral from A to B along curve C and so for evaluating this what I'm going to do in the integral I'm going to put a C showing that we're integrating along a curve and we're going from A to B and we would have our vector A and the way that we do this is we use our dot product. That would give us the vector component along the direction ds and that is the way that we evaluate a line integral. Now it turns out sometimes in fluid mechanics you'll want to evaluate one of these line integrals with a closed curve so where the curve closes upon itself and you might do that for example if you're looking at quantifying the circulation about an airfoil you can measure the velocity field integrated along a closed curve that would tell you the bound circulation on that particular airfoil but if C is closed so we might have a curve that looks like this and then I'm going to draw the differential element here ds and that is a vector and again we have our vector A so if the curve is closed the way that we do this in fluid mechanics is we will close the sign in the integral it's no longer a C and we'll put a little arrow on that and that arrow would indicate the direction with which going around the curve is being viewed as being positive so that is an integral around a curve and again to evaluate it is just the dot product of our vector field that we're interested in and ds and so by convention quite often in fluid mechanics what we will do is we say counter clockwise is positive and if you have a hard time remembering that imagine yourself walking around the curve so here you are you're walking along imagine that you're walking in the area the area that you're integrating around is always to your left and so that's an easy way to remember or just remember counter clockwise is positive okay so that's the line integral that's one that we use typically for evaluating circulation but there can be other things you might look at in fluid mechanics.