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=====Mixed Partials and a Maxwell Relation (15 minutes)=====
  * During this lecture, the instructor reviewed the new definition of the forces $F_x$ and $F_y$ as partial derivatives of the potential energy. Students were then asked to work out $\left(\frac{\partial F_x}{\partial y}\right)_x$ and $\left(\frac{\partial F_y}{\partial x}\right)_y$ 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 $\left(\frac{\partial F_x}{\partial y}\right)_x$ and $\left(\frac{\partial F_y}{\partial x}\right)_y$ 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.

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