Given some state function $$f(x,y,z)=0$$ I want to prove the reciprocity relationship $$\left(\dfrac{\partial x}{\partial y}\right)_z=\dfrac{1}{\left(\dfrac{\partial y}{\partial x}\right)_z}$$

I know how to do this the standard way (implicit function theorem, $f$ is exactly differentiable, etc.) but I've recently seen someone employ a somewhat trivial method of demonstrating this and I can't help but feel they've done something smelly but I lack the background to make a determination.

Beginning with the assumption that $z$ is held constant,

$$\dfrac{\partial x}{\partial y}=\dfrac{\partial x}{\partial y}\cdot\dfrac{\partial x}{\partial x}=\dfrac{\partial x}{\partial y}\cdot\dfrac{1}{\dfrac{\partial x}{\partial x}}=\dfrac{1}{\dfrac{\partial y}{\partial x}}$$

Is this valid? More importantly, how is $\frac{\partial x}{\partial x}$ to be interpreted? In physics we treat derivatives and partials like fractions and do algebraic operations on them all the time. What are the problems with this sort of treatment?


The short answer is no. I have read a lot of great posts extolling the virtues (or lack thereof) of the Leibniz notation (such as the top response of How is it that treating Leibniz notation as a fraction is fundamentally incorrect but at the same time useful?).

There is a way to make the abuse of differentials rigorous known as non-standard analysis, which you can read about here. Whether or not the manipulations you have written down constitute a formal proof in the context of non-standard analysis is beyond me, since I have never bothered learning it. Even if it was, to be truly rigorous you would need to build the framework of non-standard analysis from the ground up before you could quote such a proof.

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