 So there's one last useful partial derivative identity to talk about, and that's this one called a change of constraint rule or the change of direction rule. It often happens that we may know the value of some derivative of a function with respect to x at constant y, but perhaps what we're really interested in is the derivative of that same function with respect to the same variable while holding something else constant. So we often need to be able to convert back and forth between those two, how to convert between df dx at constant y and df dx at some different constant variable z. So to understand how we can do that, let's start with thinking of f as a function of x and y. If I write the differential of that function, it's going to be df dx times dx, and df dy times dy. Now what I'll do to make some progress toward making it look like this is I'll take both sides of this equation and I'll take the differentials divided by dx while doing that at constant z. So I'll say df dx at constant z is equal to the df dx at constant y that was already here and now the dx becomes dx dx at constant z. The second term looks like the df dy at constant x that was already here and now the dy becomes dy dx at constant z. So here, here and here I've taken 1 over dx at constant z. One of these derivatives we know the value, the answer to, dx dx at constant z as x changes by the exact same amount, so that derivative is equal to 1. So this simplifies a little bit and I can say df dx at constant z is equal to df dx at constant y multiplied by 1 plus df dy at constant x, dy dx at constant z. This now is exactly the thing we were looking for, what's the relationship between a particular derivative at constant y and the same derivative at constant z? That's exactly what we have here, df dx at constant z is equal to df dx at constant y plus this extra term. So if I have this one, if I also know the values of these, I can add them to it to figure out what df dx at constant z is. So I'll put that in a box to write down and refer to later. That's the rule that we can call the change of constraint rule because it tells us how to get a derivative with one constraint relative to one at a different constraint or sometimes called the change of direction rule because we're changing what direction we're taking the derivative in is at the constant y direction or the constant z direction. So as usual, this will make a lot more sense once we see how we can use it in an example. So for an example where we want to know what a derivative that we can evaluate at two different constraints, I'll remind you that we already know from the fundamental equation for du. Let's see, du is equal to tds minus pdv. So we know that du, ds at constant v, no that's not the one I want, du, dv at constant s is equal to negative pressure. So du, dv while holding s constant, so this term goes away, du, dv at constant s is equal to negative pressure. How much we know just from the fundamental equation, du, dv at constant s is not a terribly useful thing to know. We don't tend to do things isentropically while holding the entropy constant. A more common thing to do would be to do something isothermally. So it's probably more likely that we'll run across circumstances where we want to know how much the internal energy of something is changing while I change its volume holding the temperature constant. So that's one that right now we don't know anything about, but the change of constraint rule can help us figure it out. If we use that change of constraint rule to figure out du, dv at constant t, knowing what we know about du, dv at constant s, we just have to make a correspondence between these variables f. The f's are going to be like our u's, where we have x's in this change of constraint rule. We have v, where we have z's, we're going to use t, and when we have y's, the one we already know, we're going to use s. So doing that nice and carefully so I don't make a mistake, df, dx at constant y, the one we already know about, that's du, dv at constant s. So now I'm going to need df, dy at constant x, so that's derivative of f, which we're calling the internal energy. y is the entropy, x is the volume, and now dy, dx at constant z, y is entropy, x is volume, and z is temperature, alright. So far so good. du, dv at constant t is equal to the one we know, du, dv at constant s, that's negative pressure, we know that from the fundamental equation. Likewise du, ds at constant v, du, ds while holding v constant, that's the temperature. So this term, this derivative is just temperature. Lastly ds, dv at constant t, that's also one that we know something about. A Maxwell relation reminds us that ds, dv at constant t is the same as dp, dt at constant v, so that particular relationship came from a Maxwell relation. And now dp, dt at constant v, that's a derivative that we spent a fair amount of time analyzing when we discussed the cyclic rule, at which point we discovered that dp, dt at constant v, that's related to two other cyclic permutations of those same variables, and that turned out to be alpha divided by kappa, the thermal expansion coefficient divided by the isothermal compressibility. So now what we've discovered is when something changes in volume isothermally, its internal energy changes at this rate, whatever the pressure is, negative pressure plus the temperature times alpha to over kappa. So that's a very typical example of the type of thing we often do when we discover that we're interested in a particular thermodynamic relationship. We can use the partial derivative identities that we've talked about up to this point, the reciprocal rule, the chain rule, the cyclic rule, the change of constraint rule, along with things like the Maxwell relations and the named quantities like alpha over kappa. If we combine all those rules in the right ways, we can discover these identities that relate these thermodynamic relationships to things that we're already perfectly comfortable with, named quantities and basic thermodynamic variables. So that's the thing we'll do many times as we go forward to discover things about these thermodynamic relationships and to derive new ones, and then we can use those thermodynamic relationships to understand what goes on in a chemical system. So for example, the next thing we'll do is start using these thermodynamic relationships to talk about phase changes.