 So that simple f equals e minus ts equation is going to enable us to do some pretty cool things. I already mentioned things about the loss of thermodynamics and that the free energy is a local minimum at equilibrium, right? But let's look at how the free energy varies around the equilibrium. We can do that and we can do that in particular using those small differentials. So let's see what happens if I take the free energy and then I'm adding a small component that I then call df. Well that means that I first have the energy, right? So that's going to be some sort of difference in energy. And then I have t and s, that's a product and that means that I'm going to have minus t ds minus s dt. And in general I can't get much further there but now I say that we're interested in what happens at equilibrium. At equilibrium I am at a local minimum in free energy. And at the local minimum in free energy well the I'm not going to change the free energy if I'm moving the system a minimal amount, right? Otherwise I would not be at local minimum. Same thing at equilibrium, the temperature is not changing because if the temperature is changing the system would be changing. If the temperature is not changing, dt is going to be zero. The other thing that I have here, oh sorry, I should not have f there. I was just looking at the difference, right? Not the absolute values. So that means I have some that zero equals dE minus t ds. So dE minus t ds equals zero or written in another way t equals dE ds. That's pretty cool. What on earth have I done here? Well I kind of restored the order. Before this class you thought that temperature was something very simple while entropy was something very hard. I hope I managed to convince you with the large few slides that entropy is not at all difficult. Unfortunately I'm now restoring the balance by completely destroying what you think about temperature. Temperature is something exceptionally complicated. So it turns out temperature, forget about everything you know about temperature if you're looking at a thermometer. Now we're physicists, right? Based on our equations we're saying that temperature is the derivative of how energy of a system varies as I am increasing the entropy. But it's not as stupid as I think. If I have a system here and I have entropy here and then I have energy here, we're going to come back to this later. So right now I'm just going to draw a curve. This curve somehow measures as we're gradually increasing the disorder in the system. Something happens with energy. This curve could have other shapes too. And the temperature is really described by that line. Exactly what that means you don't have any gut feeling for you right now. That's fun because I bet that before this lecture each and every one of you had a gut feeling what what temperature was. And if you're complaining that you don't know what this is what you are really confessing is that you have no idea what temperature is anymore. Congratulations. I'm not going to be able to explain that to you today but later on hang on to our three lectures and it's going to show that we can use this to derive a few things. Don't worry too much about not having a good gut feeling for something. Unless you're a passionate legal accountant you likely don't have a good gut feeling for tax law but you probably do your tax returns every year simply by following the instructions and doing what you're supposed to do. And in our world that's the same thing I trust in your equations. I'm going to have a small confession. When I was your rates and started university I liked to feel that I was pretty smart. The problem with being smart is that you reason about things. You have a good gut feeling and that takes you pretty far. But when you're studying physics and in particular theoretical physics you get to points where your gut feeling leads you wrong. And I have to confess that that led me wrong as many others. The trick then is to see the equations not as a complication but they are a help. The equations is a shoulder to rest on when things are so complicated that you can't keep them in your head. So in particularly if you feel that equations are difficult you should do the exact opposite. Trust the equations. You don't need to have a fundamental gut feeling for equations just as you don't need to have for tax law. Follow them, rely on them. If you just take it slowly you can count on that the results you're getting are right if you just follow the laws of mathematics. It's easier for me to keep these sorts on paper than it is to keep it in my head. And I hope that's going to be the case for you too.