 Welcome back. We derived four equations pertaining to the second derivative relations and these we called M1, M2, M3, M4. Let me now say M was for Maxwell and these four relations are known as Maxwell's relations. Let us write them down once again. The first relation we obtained we using the thermal energy u that was partial of t with respect to v at constant s equal to minus partial of p with respect to s at constant v. The second relation we obtained was partial of t with respect to p at constant s was equal to partial of v with respect to s at constant p. The third equation was partial of s with respect to v at constant t was partial of p with respect to t at constant v. And the fourth equation was partial of s with respect to p at constant t was equal to minus partial of v with respect to t at constant p. These four relations are known as Maxwell's relations. These relations are funny. They link some property relation to some other property relation. Sometimes surprising relation which we generally will not imagine. These relations help us find our way through many property relations which we seek. And in particular these two relations there is no particular name or order for these relations. These two relations the third and fourth in our list are something special on the right hand side here as well as here. We have only p v t as the variables. So the right hand side of these equations depends only on the equation of state which is the basic equation of state is a relation between pressure, volume and temperature. So these two relations help us relate the variation of entropy with volume at constant temperature or entropy at pressure at constant temperature purely to the p v t or equation of state data. We will find these equations very useful. Now Maxwell's relations is the name given not only to these four relations but even their reciprocals. For example you take this equation and using the partial derivative relation that partial of x with respect to y at constant z equals reciprocal of partial of y with respect to x at constant z. It is a simple reciprocity relation. Conditions are required for this. For example the functions involved need to be continuous and the derivative should exist. But these conditions are almost invariably satisfied by the relations between thermodynamic properties. So generally we will use these purely mathematical identities to be applicable in thermodynamics. So using this for example this equation will become partial of p with respect to s at constant t is minus partial of t with respect to v at constant p. And similarly you can write the reciprocal of this equation, this equation and this equation. And even those equations are generally called Maxwell's relations. Since the Maxwell's relations are useful there is a need to remember them. But if you look at them there does not seem to be any straight forward structure. But since they come out of thermodynamics some simple thermodynamic principles must be involved. But before we come to that we will take a mathematical detour to be comfortable with partial differentiation, partial derivatives and some relations between them. Thank you.