 Welcome back. Now we are going to look at the derivatives of these four energy functions and the relations which we extract out of these derivatives. The four energy functions are thermal energy U, enthalpy H, Helmholtz function A and Gibbs function G. And when we work with them, we are going to use the basic property related to exact differentials and partial derivatives. For example, let us look at some math actually calculus. Whenever we take a differential of any of these properties, we are going to get a function like or a situation like dz is some m dx plus n dy. For example, Tds minus Pdv or Tds plus Vdp, some combination. This will be the typical form. In our case, z, x and y will be properties and so will m and nb. And because z and all other functions which come into here are properties, their differentials are exact differentials because z is a proper function of x and y. And because of that, this is going to be an exact differential. Mind you, any expression which looks like this need not be an exact differential. Only under certain conditions, this will be exact differential. In thermodynamics, if z is a property and if we have a proper relation which comes out in this form, then we are sure that the right hand side is an exact differential. And because this is an exact differential, what we are going to have is m will represent partial of z with respect to x at constant y and n is going to represent partial of z with respect to y at constant x. That often in elementary treatments of mathematics, we do not write this function which is maintained constant y here and x here. That is because x and y are supposed to be the common symbols. If you write partial of z with respect to x, it is by default assume that the independent variable y is kept constant. But in thermodynamics, the situation is different. Since we are considering properties of a simple compressible system, it is a simple system, so two properties are needed. The choice of the two properties is with us. So suppose I say partial of u with respect to pressure. Now the question is, well, you have considered variation with pressure. But what is the other property that you are considering when you consider u as a function of two properties? You could be considered as a function of p and v or p and t or p and s. And then you have to specify that variable which is the other variable and which is not directly seen in the partial derivative as the constant variable. So we will have partial of u with respect to p at constant v, partial of u with respect to p at constant t, partial derivative of u with respect to p at say constant s. Continuing with this, if this on the right hand side is an exact differential, then m must be partial of z with respect to x at constant y, n must be partial of z with respect to y at constant x and the crossed second partial derivatives must be independent of the order of differentiation. That means whether you take the second partial derivative of z, first with respect to x at constant y and then with respect to y at constant x. This should equal the second derivative of z first with respect to y and then with respect to x. Actually the right hand side should properly be written like this. First you take the partial of z with respect to y at constant x and then you take the second derivative that means derivative of this first derivative with respect to x and this will be at constant y. Similarly on the left hand side the the proper way of writing this would be first take the partial derivative of z with respect to x at constant y and then take the partial derivative of this with respect to y at constant x. Anyway coming back to this, when you look at this or its proper forms, you will notice that the first one here represents partial of m with respect to y at constant x and this is partial of n with respect to x at constant y. These are the most important derivations from calculus of partial differentials and exact differentials that we are going to use. Thank you.