 Welcome back. Let us devote this particular session only to mathematics. So, here you will see x, y, p, q, u, v, all sorts of symbols. The symbols have no thermodynamic meaning. For example, if you see p, do not imagine pressure. If you see x, do not imagine trinus fraction, u and v are not necessarily thermal energy and volume. They are just mathematical symbols. So, let us say that we have a functional relationship between three variables. For example, a relationship between p, v and t is by the equation of state. For an ideal gas, it is p, v equals r, t. We can either write it as p is a function of v and t or v is a function of p and t or t is a function of p and v. But we can also write it as some function of p, v and t is 0. Instead of writing it as p, v equals r, t, we can write it as p, v minus r, t equals 0. So, we have some function of p, v and t equal to 0. So, let us begin our mathematical journey. A general relationship between three variables, which is very common in thermodynamics, can be written down in a general form like this. Of course, if you need, you can rewrite it as x as a function of y and z or y as a function of x and z or z as a function of x and y. The first relationship that we are going to look at is among partial derivatives. And one of the important relationship that we are not going to derive, but note and use, is the cyclic relation. Let us consider partial of x with respect to y at constant z. Multiply this by partial of y with respect to z at constant x. And further multiply this by partial of z with respect to x at constant y. Notice here, we have x, y, z, y, z, x, z, x, y. The variation is cyclic or you have here x, y, z, y, z, x, z, x, y. So, what is the value of this product? The value of this product turns out to be minus 1. The second cyclic relation. The second cyclic relation is, suppose we consider all these three x, y and z to be written in terms of some other function or a parameter t. In that case, we can consider another cyclic relation. Partial of x with respect to y at some parameter t, then partial of y with respect to z with respect to the same parameter t and partial of z with respect to x with the same parameter t is plus 1. Notice that since we have a common thing here, it becomes plus 1, but if the variable which is kept constant during a partial derivative also varies cyclically, then we have a minus 1 on the right hand side. The third relation which we mentioned in the previous session is the simple reciprocity relation. That is, partial of x with respect to y at constant z is the reciprocal of partial of y with respect to x at constant z. And using these, we can manipulate each one of these to get this relation. For example, partial of x with respect to y at constant z is going to be negative of partial of x with respect to z at constant y into partial of z with respect to y at constant x. Whereas, the second relation will give us partial of x with respect to y at constant t equal to partial of x with respect to z at constant t into partial of z with respect to y at constant t. Now, the next important thing is a combination of partial derivatives. That is a very important combination and that is known as the Jacobian. Let us look at what the Jacobian is and what is its use. The Jacobian is like a derivative, but it is some sort of a derivative of two functions. Each one of them is a function of the same pair of variables. So, let us say the Jacobian of u v with respect to x y. This means we have two functions u and v. Each one of them is a function of x and y. Then the Jacobian is a symbol, looks like a partial derivative. Jacobian of u v with respect to x y. This is the symbol used. This is defined as the determinant of partial of u with respect to x at constant y. That is the first term in the first row, first column first row. The second column first row is partial of u with respect to y at constant x. It is a 2 by 2 determinant. The second row consists of partial of v with respect to x at constant y and partial of v with respect to y at constant x. The importance of Jacobian is that it can almost always be treated as an ordinary derivative. For example, the following identities straight away apply. Jacobian of u v with respect to x and y is the reciprocal of the Jacobian of x y with respect to u and v. That is the first condition. The second relation is Jacobian of u and v with respect to x and y, multiplied by Jacobian of x and y with respect to p and q. That means in the first term we are considering each one of u and v to be a function of x and y individually. In the second term, each one of x and y is considered a function of p and q. And this product equals the Jacobian of u v with respect to p, q. It is as if we are cancelling this term and this term in a simple product of two ratios. Now the importance of Jacobian does not stop here. In fact, it turns out that simple partial derivatives are also hidden inside Jacobians. For example, this is one important set of relation. If I have a Jacobian where one of the functions in the numerator and denominator is common. For example, let us consider Jacobian of x and u with respect to y and u. Now we are considering x to be a function of y and u and u to be a function of y and u, but since u is u itself, the u as a function of y and u will be u itself, a simplified function. This will equal partial of or Jacobian of u x with respect to u y. This equals negative of the Jacobian of x and u with respect to u and y, which equals negative of the Jacobian of u and x with respect to y and u. That means the value of a Jacobian does not change if you flip the terms for the variables in the numerator as well as the denominator together. But if you flip only the pair in the denominator or the pair in the numerator, then the value changes its sign. The most important thing is each one of this is equal to partial of x with respect to y at constant u. Such variables we see or such partial derivatives we see very often in thermodynamics. We have seen them already in the number of places. So now we can write them in terms of Jacobians. Now what is the use of writing them in terms of Jacobians? Now Jacobian not only finds its place in the normal calculus, but makes important contributions even to analytical geometry. For example, let us consider a region defined by x and y. And let us say that there is some loopy figure, closed figure in this region. The area of this, let me call this area in the x, y plane. And now let us say that we define some function u of x and y, we define some function v of x and y and for every point on this figure which is represented by a pair x and y, we will find a corresponding u and v. And maybe the corresponding figure corresponding to this in the u v plane is this. As we traverse like this, let us say we traverse like this. Now let us say the area of this closed part which corresponds to this loop in the x, y plane is area in the u v plane. Now the area in the x, y plane can be written down as a double integral over this closed figure of dx dy. The area in this plane is going to be double integral again closed loop of du dv. Now how do you transform this? The transformation rule says that this will be equal to the Jacobian of x, y with respect to u v du dv. So that means that this factor which enters here is the scale factor. If the scale factor is 1, that means the areas in the two planes would be the same. If the scale factor is higher than 1, that means the area in the x, y plane will be larger. If the scale factor is lower than 1, then the area in the u v plane will be larger. We will make use of this in our thermodynamic derivations now. Thank you.