What if T, p, and n may vary? In this case, we get the total differential of U: (dr) dr + (dp) T. dU dT dT T,n p,n dU = au T dU = What are these derivatives and how to you evaluate them? This is what you will have learned if you have taken a multi-dimensional calculus course. More properly, the derivatives are called partial derivatives because you are taking a function of multiple variables (in the example U is a function of three variables) and taking derivatives with respect to just one of the variables holding the others constant. As an example, assume that U has the following dependence on the three variables: p.n U(p, T, n) cT np where c is a constant. The three derivatives needed for dU are (recall that if something is a constant it is not affected by the derivative) au Jp) T.n au n p,n = = = dp + dU dn 3cTnp 2cTnp cr3p2 p = P,T dn where the partial symbols denote partial derivatives. Note the partial derivatives are obtained by assuming all the other variables are just constants. In this case 3cT npdT+2cTnpdp + cTp dn nRT V 1. Test yourself. Write the total differential dp for a gas obeying the perfect gas law (make sure to evaluate any partial derivatives):