 Let's try to adapt our conservation of mass equation for a differential form. To begin with, let's consider a fluid volume. This control volume is arbitrary. It doesn't matter where it is, so long as it's not on a boundary. Since we have a control volume, we can apply our integral form of our conservation of mass, and then we can simplify it by collapsing a couple of our vectors. I can write the derivative with respect to time inside the integral, as it's a control volume. And then I can pop the vector off the area by writing that out as a normal, because remember, we're describing the x, y, and z components individually. So we'd be describing mass flow rates normal to each of the six faces of the cube. I can simplify this integral a little bit more by remembering my divergence theorem from calculus. The divergence theorem says the integral across this volume gradient of g vector is going to be the cyclic integral across an area, the g vector normal, the area. Therefore, I can write my integral above, now that these are both integrals with respect to volume, I can combine them, and then I can recognize that for any control volume, the only way that that integral can evaluate to zero is if the thing being integrated itself is zero. Therefore, the quantity inside the integrand must be zero. In a Cartesian coordinate system, therefore, my conservation of mass becomes derivative of density with respect to time, derivative with respect to x, density times u, derivative with respect to y, density times v, derivative with respect to z of density times w, and that entire quantity is zero. For polar coordinates, I could write that as partial derivative of density with respect to time plus one over r, derivative with respect to r, and then r, density, v, r plus one over r, velocity in the theta direction plus derivative with respect to z. In fact, I can group these into some simplifications around this quantity and organize this a little bit more clearly. Let's take conservation of mass, and then we can simplify it for a couple of situations that we encounter frequently for steady state analysis nothing can change with respect to time. So that would just become partial derivative with respect to x, density times u plus partial derivative with respect to y, density times v plus partial derivative with respect to z of density times w is equal to zero. Or we could simplify that a little bit further by writing that as a gradient, gradient times density velocity vector itself is zero, a little bit more convenient to write. Or for incompressible flow, at which point I would just have w, del x, del v, del y, plus del w, del z is zero, which can be more conveniently written as the gradient of the velocity vector is zero. Let's try an example.