In thermodynamics, partial derivatives state which variable is held constant. For example $\frac{\partial U}{\partial V}\vert_T$ means the partial derivative of the internal energy $U$ with respect to the volume $V$, keeping the temperature $T$ constant.

What happens to the other variables in the function $U$ if they are not held constant? As an example, what would $\frac{\partial f}{\partial x}\vert_y$ be if $$f(x,y,z) = x^2 + 5y + z^3$$

is it $$\frac{\partial f}{\partial x}\vert_y = 2x + 3z^2\frac{d z}{dx}$$

or something else?

  • $\begingroup$ Your guess is sort of correct, if $z = z(x)$ then what you wrote is correct but if $z = z(x, ...)$ then you need to write $\frac{\partial z}{\partial x}$ instead of $\frac{d z}{d x}$ and it needs to be evident what variables are being held constant in the partial derivative. This can be done by writing $\left( \frac{\partial z}{\partial x} \right)_{ab}$ where or just by writing out $z$ as an explicit function like $z(x, a, b) = x^2 + x a + b$ as the case may be. $\endgroup$ – jcarpenter2 May 2 '20 at 14:21

All variables other than $x$ in your example must be held constant. The ambiguity arises from definitions like

$$f = x + y = 2 x + z$$

Here it is evident that $y = x + z$ but it is not specified whether $f$ is a function of $(x, y)$ or $(x, z)$ if you were to write $\frac{\partial f}{\partial x}$. Since there is nothing to say which is the more fundamental quantity, $y$ or $z$ (or if it even makes sense to ask that question), "there is nowhere to stand" (as the Buddhists say), and you must specify which variable, $y$ or $z$, you are choosing to hold constant.

$$\left( \frac{\partial f}{\partial x} \right)_y = 1$$

$$\left( \frac{\partial f}{\partial x} \right)_z = 2$$

If some variables are not held constant, the partial derivative is not well defined in the first place, since ambiguities like the above can be created by rewriting any variable in the function. To take a derivative the function must be a function only of the variable with respect to which it is being differentiated.


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