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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.)

Fll'Yissetat
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Alex Gavrilov
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    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 $v\mapsto v^*$. – Piotr Achinger Feb 09 '21 at 08:01
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    (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 $\partial_i f$. – Piotr Achinger Feb 09 '21 at 08:02
  • https://hsm.stackexchange.com/ is a right forum for such questions. – user64494 Feb 09 '21 at 08:28
  • @ Piotr Achinger Yes, but the whole point is to not say anything! – Alex Gavrilov Feb 09 '21 at 08:28
  • Is there something wrong with $\frac{\partial f(x,y)}{\partial x}$ from your first link? – Najib Idrissi Feb 09 '21 at 10:07
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    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. – Zach Teitler Feb 09 '21 at 10:39
  • What is wrong with the usual notation for Lie derivatives from differential geometry: L_v f, where v is a vector field and f is a function? – Dmitri Pavlov Feb 09 '21 at 14:34
  • @DmitriPavlov: When you say what vector field $v$ you are using, if you work in coordinates, you need to write it in a basis of coordinate vector fields. You have the same problem as above if you use two systems of coordinates, where the same function $x$ is one of the coordinates in both systems. – Ben McKay Feb 09 '21 at 15:01
  • @NajibIdrissi in the context of the question you would need to write $\frac{\partial f(u,v)}{\partial u}$, which would abuse application notation $f(x,y)$, since $(u,v)$ are not elements of $\mathbb{R}^2$, while $f$ is a map from $\mathbb{R}^2$. – Michael Bächtold Feb 11 '21 at 07:48
  • I had never thought of, and love (though I recognise its unsuitability for many purposes, e.g., pedagogy), the notation $\frac{df \wedge dv}{du \wedge dv}$! – LSpice May 19 '21 at 20:48

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