## Mixed Partials and a Maxwell Relation (15 minutes)

- During this lecture, the instructor reviewed the new definition of the forces $F_1$ and $F_2$ as partial derivatives of the potential energy. Students were then asked to work out $\left(\frac{\partial F_1}{\partial x_2}\right)_{x_1}$ and $\left(\frac{\partial F_2}{\partial x_1}\right)_{x_2}$ as derivatives of the potential energy, $U$.

- Introducing Clairut's Theorem, the order in which you take partial derivatives does not matter, the instructor then helped students find that the derivatives $\left(\frac{\partial F_1}{\partial x_2}\right)_{x_1}$ and $\left(\frac{\partial F_2}{\partial x_1}\right)_{x_2}$ were equivalent.

- Since both of these derivatives fall in the category of “hard” derivatives (they cannot be easily measured with a PDM) students were told that some modifications of a PDM (such as the inclusion of force meters in the strings) would be necessary to confirm the equivalence of these two derivatives.