 So we're going to continue looking at kinematic variables using the velocity equation and we just looked at volumetric flow rate. We're going to end by looking at three different ones but we'll begin with mass flow rate and once you determine volumetric flow rate, mass flow rate is relatively simple. In this course quite often we'll refer to that as being m with an over dot or you'll hear m dot and that will be mass flow rate. So mass flow rate or m dot is the integral over a surface integral again. So look back to the segment that we just had on volumetric flow rate but the only difference here is that we put the density into the integral so we have the dot product of the velocity multiplied by the unit vector which is normal to the surface and then integrated over the surface of the object that we're looking at or of the body. Now it turns out that if rho is equal to a constant then what you can do is you can pull the density out of the integral and you would have something that looks like this and if you recall back from the segment that we just looked at where we evaluate the volumetric flow rate that means that we can write it in this way m dot would then be rho times q. So that is mass flow rate. The next one that we will look at is rate of volume expansion per unit initial volume and so with this one what we have is 1 over v and quite often in fluid mechanics you'll notice we'll use a v with a line through it to denote volume that would be rate of volume change expansion or contraction with respect to time per unit initials that's why we multiply it by 1 over v. Now in fluid mechanics we can express this with an operator del dot and del dot is the divergence and it is the divergence of the velocity field. So divergence which is there or sometimes you'll hear it here it referred to as being del dot and we can write that out that's a dot product. So del dot is expressed as that in terms of partial derivatives of our three velocity components uvw and if it turns out that we have a flow where del dot v is equal to zero this would be the case where the volume does not expand or contract in the fluid flow and that is what we would call incompressible. So if you're dealing with a fluid mechanic problem and they say the flow is incompressible that tells you immediately that you're dealing with something where del dot v is equal to zero that is for an incompressible flow and so that is sometimes something that you'll need to remember when you're dealing with incompressible problems but that is the rate of volume expansion previously we looked at mass flow rate the final thing we're going to look at here is going to be the angular velocity and angular velocity is another thing that's important in fluid mechanics it's related to the vorticity if you recall from an earlier segment where I talked to you about why study fluid mechanics we looked at the inlet to an engine triple seven engine you saw swirl flow angular velocity quantifies vorticity there'd be a lot of vorticity in the region where you had that vortex but the way that we define angular velocity we do that by determining the velocity field to begin with and we use the symbol omega with the vector and that is equal to one half del cross v and this is also sometimes referred to as being the curl of v so if you hear del cross v or curl of v that refers to this operator here and we can expand that out I j k and then the second row we have partial partial x partial partial y partial partial z and then the third we have u v w and so with that that would be the way to determine the angular velocity which again is related to vorticity in a fluid flow now if it turns out if del cross v is equal to zero that is an indication of a flow a special kind of flow that we call irrotational and if it turns out that if we have a flow where del dot v is equal to zero and del cross v is equal to zero that would be a flow that is incompressible and irrotational and that is a special subset of fluid mechanics that we refer to as being potential flow and we will be looking at that later in this course you can do some interesting things with potential flow but it only applies if we have these two requirements del dot v equals zero and del cross v is equal to zero so those are some things that you can do with velocity we looked at acceleration we looked at volumetric flow rate we looked at mass flow rate we looked at volumetric expansion and then finally we've looked at angular velocity so those are things that you can do with the velocity vector field and we'll be using those throughout this course