 When we describe fluids with integral relations, we are primarily using a Lagrangian perspective. We are looking at a large control volume and considering its contents at any given moment. In some circumstances, it is more useful to describe a fluid from a Eulerian perspective, where we follow a single packet of fluid. That process of considering a packet of fluid, be it an infinitesimally small control volume or a unit mass, allows us to develop relationships on a small scale, a differential analysis from which we build differential equations. Within our differential relations, we will need to refer to our fluid properties from the same perspective. We need to be able to describe what's happening to our packet instantaneously. The easiest way for us to build these relations is to start from a Lagrangian point of view and adapt it to a Eulerian point of view. One such example of this shift in perspective is representing a material derivative in terms of Eulerian quantities. A material derivative is the time derivative, the rate of change, of a property following a fluid particle. Let's consider the material derivative of velocity. We typically represent velocity as a function of space and time, which in Cartesian coordinates would be a function of x, y, z and time. Then the derivative of velocity with respect to time would have to include each of those independent variables. It would begin as a partial derivative of velocity with respect to time. And then we add to that the partial derivative of the velocity vector with respect to x, and then the derivative of x with respect to time, plus the partial derivative of velocity with respect to y, and then the derivative of y with respect to time, and then the partial derivative of velocity with respect to z, and then the derivative of z with respect to time, still a partial derivative. And then we can recognize that for convenience, we often refer to the x, y, and z components of velocity as their own quantity. We described them as u, v, and w for Cartesian coordinates. As a result, we can make the substitution and write this as the partial derivative of velocity with respect to time plus u times the partial derivative of velocity with respect to x plus v times the partial derivative of velocity with respect to y plus w times the partial derivative of velocity with respect to z. Then we can pull u, v, and w out and write that as a gradient of the velocity vector itself, at which point this would become partial derivative of v with respect to time plus velocity vector gradient velocity vector. In this acceleration, this first term is what we call the local acceleration, and then the term on the right is what we call convective acceleration. That process of writing out the derivative of the velocity with respect to time can be abbreviated as an uppercase d, and that uppercase d means that we're writing this as partial derivative of the velocity vector with respect to time plus the velocity vector times the gradient of this velocity vector, and that can apply to anything by which I mean any macroscopic tensor field. We could write out d anything, and then that material derivative would become the partial derivative of that quantity with respect to time plus the velocity vector gradient that quantity. So for example, I could write that out for pressure. The material derivative of pressure with respect to time would be the partial derivative of pressure with respect to time plus the velocity vector gradient pressure, and what that represents is the partial derivative of pressure with respect to time plus u times the partial derivative of pressure with respect to x plus v times the partial derivative of pressure with respect to y plus w times the partial derivative of pressure with respect to z.