 So we just talked about the velocity field of a fluid. Now what we're going to do, we're going to look at another property and that is acceleration, which is something that you get from the velocity field. So acceleration is also something that is very important because it's in Newton's second law, but let's take a look at acceleration. So we're writing out the velocity vector that we talked about in the last segment. Remember we said u is for the x component, little v is for the y component, and then w is for the z-axis or z component, which is multiplied by unit vector, little k. Now when we look at acceleration, it is expressed as being the derivative of the velocity vector. Remember the velocity vector is a function of four variables, the three spatial and time, with respect to time. And when we write this out, we get a formulation like this using the chain rule of calculus. And we're using partial derivatives here because velocity is a function of more than one variable. So with this, and that was by the chain rule that we're able to obtain that. Now when we look at this, we'll notice that we have these terms here, dx, dt, dy, dt, dz or dz, dt. Those are just the three velocity components. So we can write out little u is dx by dt, little v is dy by dt, and finally little w is dz, dt. So what we can do, we can rewrite the velocity, or sorry, the acceleration in two different components. We have the velocity vector with respect to time, and then we have this other section or component, which consists of the velocities. And we will refer to the first term here as being the local acceleration. That is at a given point. And then the second one is referred to as being the convective acceleration. Now it turns out if the flow is steady, if the flow is steady, that means that it is not changing with respect to time. And consequently in that case, we can write dv by dt is equal to zero. So actually the first term, the local acceleration term would disappear if we were dealing with a steady flow. So what we're going to do, we're going to take a look at rearranging this expression for the acceleration of a fluid. And so let's take a look at rearranging that mathematically. So this is referred to, let's take a look at the second term. That was the convective acceleration. And we are going to rewrite it using some of the different operators that you learn in your mathematics courses. And the one that we will use is the gradient operator. And if you recall the symbol for the gradient operator is an upside down triangle like that. So the gradient operator is defined as being partial by partial x in the i direction plus partial by partial y in the j direction plus partial by partial z in the k direction. So when we look at our convective acceleration, so remember the convective acceleration, let's go back and look at it. That was this term in our acceleration. This is our convective acceleration. We're going to rewrite that now using the gradient operator. So that was our convective acceleration. And we can then re-express that using the gradient operator as this. We take our velocity dotted into the gradient operators, that would be v dot del. And it is then multiplied by the velocity vector v. Therefore, we can rewrite the acceleration as being our local acceleration, which was dv by dt. So this would be for a generic flow. It could be steady, it could be unsteady. It would be unsteady and we would keep that term. Plus then what we just wrote using the gradient operator v dot del multiplied by the velocity vector v. So this is a form of the acceleration vector in a fluid. And it's kind of an important one. But what we'll do, we'll take a look at this one step further. So although we've written this for the acceleration, it turns out that we can use this for any fluid property, be it a vector or a scalar. And what we just looked at, we applied this to a vector and obtained the acceleration. But we can also use the same operator. And we can apply it to any of the properties scalar or vector. And so we will express this as a little d, let me change color, little d with respect to t. And I'm going to draw it with respect to write it out some arbitrary property so it could be a scalar or a vector. And that would be equal to the partial with respect to time plus v dot del multiplied by whatever we're dealing with, be it a vector or a scalar. And this is a very important derivative operator that we have within fluid mechanics. And it's important that we give it, in order to denote it, we denote it with this capital D. And this is referred to as being the substantial or material derivative. And written generically here, it could apply to either a scalar or a vector. So that is the substantial derivative or the material derivative. So, for example, let's say you're wanting to compute the value for pressure. And so if you're looking to compute the value for pressure, that would be applying the substantial or material derivative to the scalar field pressure. So the unique thing about the material derivative, so the unique thing about the material derivative that we just looked at, is that it follows a particle of fixed identity. And therefore, it makes it possible for us to apply the laws of particle mechanics in the fixed Eulerian flow field frame that we talked about earlier. And so material derivative is something quite important that we'll see in a lot of different equations, mainly Newton's second law, F equals ma, from which a lot of the analysis will derive. And so that is the material derivative and acceleration from velocity field, a very important property within fluid mechanics.