# Proper notation for partial derivative

I'm writing some notes on differential calculus over normed spaces and I'm having some trouble finding the appropriate notation for partial derivatives. Here's what I mean:

Derivative of a map between normed spaces. If $$f:E\to F$$ is a map between normed spaces $$E$$ and $$F$$ over the same field $$\mathbb{K}$$, I say it is differentiable at $$a\in E$$ if there exists a continuous linear map $$T\in\mathcal{L}(E,F)$$ such that $$\lim_{h\to 0}\frac{\lVert{f(a+h)-f(a)-Th}\rVert_F}{\lVert{h}\rVert_E}=0.$$ It is easy to see that if $$T$$ exists then it is unique. I call it the derivative of $$f$$ at $$a$$, and denote it by $$\mathrm{d}_a f$$. The reason I choose to use this notation and not the (more common) $$\mathrm{D}f(a)$$ is that, since the derivative is itself a map, it can take inputs $$v\in E$$, so in the second case I would have to write the more awkward $$\mathrm{D}f(a)(v)$$ instead of the more comfortable $$\mathrm{d}_a f(v)$$ for the image of $$v$$ under $$\mathrm{d}_a f$$. See this answer from Qiaochu Yuan for more context.

Partial derivatives. Let $$E=E_1\times\cdots\times E_n$$ be a product of normed spaces and let $$F$$ be another normed space. Let $$f:E_1\times\cdots\times E_n\to F$$ be a map, which can be written as $$(x_1,\ldots,x_n)\mapsto f(x_1,\ldots,x_n)$$, where $$x_i\in E_i$$ for $$i=1,\ldots,n$$. Let $$a=(a_1,\ldots,a_n)\in E$$ be fixed. If the map \begin{align*} E_r&\to F\\ x_r&\mapsto f(a_1,\ldots,a_{r-1},x_r,a_{r+1},\ldots,a_n) \end{align*} is differentiable, I say that its partial derivative with respect to to the $$r$$th variable at the point $$a$$ is the derivative of the above map at the point $$a_r$$.

Question. As I said, I can't decide what notation to use here for partial derivatives. Some authors use $$\mathrm{D}_r f(a)$$ or $$\partial_r f(a)$$ to denote it, but we run into the same "problem" I mentioned before (the partial derivatives here are itself linear maps). The most comfortable notation I have found so far is $$\partial_r f\rvert_a$$, but it still does not convince me, because I want to have a very similar notation for the case $$E_1=\cdots=E_n=\mathbb K$$, where we get the classic partial derivative by identifying the derivative with its value at $$1\in\mathbb K$$: $$\partial_r f(a) = (\partial f/\partial x_r)(a) = \partial_r f\rvert_a (1)\in F$$.

Any help or suggestion is very appreciated.

• I’m struggling to think of anything better than $\partial_r f(a)$. Maybe just resign yourself to that. I think the notation $Df(a)(v)$ is not so bad. Mar 6, 2023 at 3:00
• You might consider the notation suggested in Fite's paper - "Total and Partial Differentials as Algebraically Manipulable Entities" arxiv.org/abs/2210.07958 Mar 6, 2023 at 3:17
• I usually put the points of evaluation in subscript position, but sometimes if I’m tired I would write $\frac{\partial f}{\partial x^i}(a)\cdot\xi$ or $\frac{\partial f}{\partial x^i}\bigg|_a\cdot\xi$, with the $\cdot$ denoting evaluation of a linear map on a vector, because in the case of $\xi\in\Bbb{F}$, this is literally equal to the numerical partial derivative (the value of the linear map at $1$) times the scalar $\xi$. Other times, if I don’t want to introduce the Leibniz notation, then I’d write $(D_jf_a)(\xi)$, with or without more brackets depending on how much clarity is needed. Mar 8, 2023 at 7:13
• it is impossible to be consistent 100% of the time; you’ll get tired eventually and know what you’re talking about and drop the fully precise notation. But till you get to that stage, just pick something, and stick with it. Mar 8, 2023 at 7:14
• Thank you all. I will take all your suggestions into account. Mar 8, 2023 at 11:20

There are a few standard notations for partial derivatives of maps between normed spaces:

Df(a)(v) or Df(a).v for the total derivative at a, as a linear map. ∂rf(a) or ∂f/∂xr(a) for the partial derivative with respect to the rth variable. ∂rf|a to emphasize that this is the derivative of the "restricted" map xr ↦ f(a1, ..., ar−1, xr, ar+1, ..., an).

So for your example, I would write: ∂f/∂xr(a) = ∂rf|a(1)

The notation ∂rf|a has the advantage that it's consistent whether xr ranges over K or a general normed space Er. The notation ∂f/∂xr(a) is more standard when xr ranges over a scalar field (like R or C), but is a bit inconsistent when Er is a normed space.

Either notation is fine, so I would choose based on what you're emphasizing or what your audience would find most familiar.

• Please use MathJax to format equations.
– Gary
Mar 6, 2023 at 4:08