 Hello, this is Professor Steven Nesheba, and I just want to tell you a little bit about these tools for thermodynamic proofs. So the context here is that we're looking at a differential equation of state that looks like this. This is the state function, and these are the state variables x and y. They could be pressure and temperature. One thing just to note is that when we write it this way, then what we mean is that this coefficient, A is the slope of F in the x direction while holding y constant, which is what I kind of drawn here. This dot means that slope in that direction, and similarly with B, that's the slope of F in the y direction while we hold x constant. So there's just a few rules. One of them is the no-brainer, which says that if I try to change x, look at how x changes with respect to y while I hold x constant, well, that's zero. And if I try to look at how x changes with respect to x while holding anything constant, well, that's just one. There's an inverter, which just says the slope of x with respect to y holding z constant is one over the looking at it in the other direction, y as a function of x. There's this thing called the converter to algebraic form, which goes something like this. If I have this differential equation of state, then and if I want to convert it to an algebraic form, let's say I want to look at F with respect to z while holding w constant. The point is that you can do that on the left-hand side and you have to do it to everything on the right-hand side. The A's and B's stay the same. There's still those coefficients, but we have to do the same thing to x as we did to F, namely partial with respect to z holding w constant and same thing for y. Another idea has to do with the Euler relation, which in our context is going to be really, we're thinking about contours of F. So going back here, you know, this might be a contour, that is to say walking along a path at which F stays constant and this rule goes like this. The slope of x with respect to y along such a contour, which in this case would be the slope of x as a function of y, is equal to just the negative of those two slope coefficients, minus B over A, which are written this way as well. Another one is what we call the product rule. It just says that if I'm looking at the small change in a product PV, that's P times the change in volume plus B times the change in pressure. That could have been any any dynamical variables. And then we have what's called the total differential. It converts the algebraic forms to differential ones. For example, if I know that h is equal to u plus PV, the differential form for that would be take a small change in h and it must be equal to a small change in u plus a small change in PV. And this last equation here, I just apply the product rule right above there.