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This question is related to Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix? and Suggestions for good notation .

When there are two functions $f,u:\mathbb{R}^2\to \mathbb{R}$, the partial derivative $\frac{\partial f}{\partial u}$ does not quite make sense. When there are three $f,u,v:\mathbb{R}^2\to \mathbb{R}$ it does, except in this case it depends on the pair $(u,v)$ of "independent variables" and omitting the second one from the notation is potentially problematic.

So, how do you best denote a partial derivative in this situation? The fraction $\frac{df\wedge dv}{du\wedge dv}$ is mathematically flawless but cumbersome, especially if there are many variables. The notation common in physics (in statistical mechanics in particular) is $\left(\frac{\partial f}{\partial u}\right)_v$, but for some reason I do not recall seeing it in any mathematical text. (Besides, it becomes really confusing if you mistake $v$ for something other than a function.) I am interested in suggestions, examples from the literature, pros and cons.

(I do not mind if this is community wiki.)

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    $\begingroup$ Let me say something obvious. This ambiguity appears every time there is a dual basis: if $v_1, \ldots, v_n$ is a basis of $V$, we usually denote the dual basis of $V^*$ by $v_1^*, \ldots, v_n^*$, which makes it seem that $v_1^*$ depends only on $v_1$ and that there is some operation $v\mapsto v^*$. $\endgroup$
    – Piotr Achinger
    Commented Feb 9, 2021 at 8:01
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    $\begingroup$ (Continued) In the problem at hand, one could simply say "let $u_1, \ldots, u_n\colon U\to \mathbf{R}$ be a local coordinate system," and let $\partial_1, \ldots, \partial_n$ (or $\partial/\partial u_i$ but that raises the same issue) be the basis of the tangent bundle dual to the basis $du_1, \ldots, du_n$ of the cotangent bundle. Then the partial derivative is $\partial_i f$. $\endgroup$
    – Piotr Achinger
    Commented Feb 9, 2021 at 8:02
  • $\begingroup$ hsm.stackexchange.com is a right forum for such questions. $\endgroup$
    – user64494
    Commented Feb 9, 2021 at 8:28
  • $\begingroup$ @ Piotr Achinger Yes, but the whole point is to not say anything! $\endgroup$
    – Alex Gavrilov
    Commented Feb 9, 2021 at 8:28
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    $\begingroup$ There seem to be two choices. Either pick a single choice of coordinates and stick with it forever. Or else use notation that fully describes which coordinate chart you're using at each moment. $\endgroup$
    – Zach Teitler
    Commented Feb 9, 2021 at 10:39

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