 Another vector operator that we use often in fluid mechanics is that of the gradient of a scalar field. And what the gradient operator does is it takes a scalar and it converts it into a vector. So we're going to begin by considering a scalar field. One that we often use in fluid mechanics is pressure. If this was heat transfer, it might be temperature. But we're going to work with pressure now. So we'll consider a scalar field. And with the scalar field, I give it a P and did an old pressure. But like I said, it could be temperature. So if we're looking at the gradient of P, and we'll define that as being grad P, I should put that in here, grad P is what you'll hear people refer to it as. Okay. And so if you take the gradient of this scalar field P, what we'll find is that the resultant of the gradient, the magnitude of the gradient that is, it's a vector and grad P will become a vector. But the magnitude of that vector is the maximum rate of change of P per unit length in that coordinate space. And the second thing is that the direction of that new vector grad P is the direction of the maximum rate of change. So it's the direction of maximum rate of change. And the magnitude is the maximum rate of change of that scalar in this coordinate system that we're dealing with through space. So let's take a look at that. And we'll draw a little picture to help illustrate what we're talking about. But beginning, we have grad P. And mathematically, it is defined in the following manner. And so with that, we're converting our scalar into a vector. And let's draw out a picture. So here we have our coordinate system. And what we're going to do, we're going to assume that we have iso lines. So lines of constant pressure in this field. So we'll have P1 equals a constant. And we'll have another line that would be P2. And finally, we'll have another line up here that would be P3. And these are iso lines of pressure. For dealing with temperature, those would be isotherms. Theoretically, it should be isobar if we're constant pressure. And we're going to assume that P3 is greater than P2 is greater than P1. And also on here, what we have, we can sketch out what I'll call gradient lines. And gradient lines are going to be perpendicular, everywhere perpendicular to the iso lines. So the gradient lines are going to do something like this. And so every place they're normal to our iso lines. And so those there are gradient lines. And then finally, what I'm going to do is we're going to take some point x, y, maybe I'll put it down here. So some point of interest. And if we were to compute the gradient at that point, and then draw the vector, it would be going in this direction here. So basically in the direction of maximum change, the magnitude would give us that maximum change in the direction tells us the direction of the maximum change. So that would be grad P. So that is an image in terms of looking physically what the gradient operator does and what it can provide us with. Now, sometimes we are after maybe not only the change in the direction of maximum change, but sometimes we might often be interested in the change in a different direction. For example, let's say you were interested in the change in some other direction here, and you wanted to know that. So what we're now going to look at is a way that would enable us to calculate change in any kind of arbitrary direction. And so again, I'm going to draw a little image or schematic in our coordinate system. And we have some point here x, y where we've evaluated the gradient. And let's say when we do that, we determine that the gradient is a vector shown there. But let's say we're interested in something different. Let's say we want to know what about this direction here. I don't want to know the gradient direction. I want to know the rate of change in this new direction s. So what we would begin by doing is determining a unit vector in direction s. So I'm going to sketch out a little unit vector here. And then what we would do is in order to determine the rate of change in direction s, that would be the gradients of p dotted into the unit vector n. So we rely on the dot product and this is the directional derivative in the s direction. And so that is sometimes what you depending upon the problem, you might want to determine that. So that is the gradient operator. Showing the gradient operator, we can write it out sometimes in this manner where we have it being applied to some generic thing that I'm putting in brackets here. So really what I should do, I should do grad and then that thing in brackets would be there. So that's the gradient operator. It tells us the maximum change and the direction of the maximum change of our scalar field and it goes from a scalar and converts it into a vector.