# proper notation for derivatives and differential

This is a simple question (perhaps pedantic) about basic differentiation in real coordinate space. In practice, I don't ever have issues using or carrying out differentiation in $$\mathbb{R}^m$$. But I'm pretty sure I frequently and severely abuse notation and I'm trying to improve.

question 1: Let $$f:\mathbb{R}^m \to \mathbb{R}^n$$ be some smooth map between real coordinate spaces. It maps some point $$x\in\mathbb{R}^m$$ to $$y\;\dot{=}\;f(x)\in\mathbb{R}^n$$. What is the proper way to write the derivative of $$f$$ (Jacobian)?

$$df \quad, \quad \frac{\partial f}{\partial x} \quad, \quad \frac{\partial f(x)}{\partial x} \quad, \quad \frac{\partial y}{\partial x} \qquad ?$$

does $$\frac{\partial f}{\partial x}$$ even mean anything or do we need to "feed" $$f$$ some input before we can differentiate as $$\frac{\partial f(x)}{\partial x}$$? In the last of the above, $$y$$ is just a point, $$y=f(x)\in\mathbb{R}^n$$, not a function, that can be expressed in terms of $$x$$. Can we differentiate a point as $$\frac{\partial y}{\partial x}$$ or is this simply a common abuse of notation?

question 2: continuing the above, what is the proper way to write the derivative of $$f$$ at a particular point, say $$p\in\mathbb{R}^m$$? which of the following are correct, incorrect, or equivalent?

$$df(p) \quad, \quad \frac{\partial f}{\partial x}\big|_p \quad, \quad \frac{\partial f(x)}{\partial x}\big|_p \quad, \quad \frac{\partial f(p)}{\partial p} \qquad?$$

I have a hunch that the "proper" notation for the derivative of some $$f:\mathbb{R}^m \to \mathbb{R}^n$$ is just $$df$$ and this is defined such that, at any arbitrary point $$x\in\mathbb{R}^m$$, it is given by $$df(x)=\frac{\partial f(x)}{\partial x}$$. Is this correct?

context: I posed this question in the context of real coordinate space(s), but I'm asking it with differential geometry in mind. My background is not in math and I recently started teaching myself some differential geometry and quickly realized I don't have a great grasp of proper mathematical notation.

edit: This question is related to one I asked on the physics page at this link

• Re. your second question, $\frac{\partial f(p)}{\partial p}$ is definitely not correct because $p$ is fixed. Related Commented Oct 11, 2022 at 17:27

$$\frac{\partial f}{\partial x}$$ always means a partial derivative, which means that $$x$$ is a specific coordinate with respect to some choice of coordinates of the domain. It's sloppy and confusing to use it to mean the total derivative and I would avoid doing that. I would write $$\boxed{ df }$$ for the total derivative and $$\boxed{ df_p }$$ for the total derivative at a point $$p$$. You can see this notation used e.g. on the Wikipedia article for total derivative.
I would avoid writing $$df(p)$$ because the total derivative is itself another function, namely a linear map, which takes as input a tangent vector $$v$$ at the point $$p$$; in my preferred notation this can be written $$df_p(v)$$ but if you want to use $$df(p)$$ you'd have to use the more awkward $$df(p)(v)$$; in my opinion this doubling of parentheses is hard to read and should be avoided.
• I confess, I use to write $\mathrm{d}f(p)v$ or $\mathrm{d}f(p)\cdot v$, thinking of $\mathrm{d}f(p)$ as an operator acting on the vector $v$. At least, I definitely do not use $\mathrm{d}f(p)(v)$. Commented Oct 11, 2022 at 16:52
• ok I think I inadvertently combined two similar questions into one by conflating partial vs total derivative. So you are saying that $df$ is the (total) derivative and it has matrix representation (in some basis) given by the jacobian $\frac{\partial f}{\partial x}$, correct? So then is $\frac{\partial f}{\partial x}$ indeed the proper way to write the jacobian or should it be $\frac{\partial f(x)}{\partial x}$? If we name $f(x)$ as $y=f(x)$, is it "proper" to write $\frac{\partial y}{\partial x}$? I.e. do we take partials of the function, $f$? or do we take partials of the output, $y=f(x)$? Commented Oct 11, 2022 at 17:13
• @JPeterson In terms of differential forms, it holds that $\mathrm{d}f = \sum_{i=1}^n \frac{\partial f}{\partial x_i} \mathrm{d}x_i$. If one writes $f=(f_1,\ldots,f_m)$, then the matrix representation of $\mathrm{d}f$ is $\left( \frac{\partial f_j}{\partial x_i}\right)_{1\leqslant i \leqslant n, 1\leqslant j \leqslant m}$. Commented Oct 11, 2022 at 17:24
• @JPeterson: again, $\frac{\partial f}{\partial x}$ is a partial derivative. I would write the Jacobian as $J(f)$ (and the Jacobian at a point as $J(f)_p$) to avoid confusion. The Jacobian is a matrix whose entries are partial derivatives, as Didier says. I would avoid the notation $\frac{\partial y}{\partial x}$ even for the partial derivative; this notation is common in physics but I think it is very confusing. Commented Oct 11, 2022 at 17:29
• Isn't $d_pf$ more comfortable? Instead of $df_p(v)$ ist would read $d_pf(v)$. That's the way I've learned it. For example, the derivative if the $k$-th coordinate function $x_k$ at a point $p$ in direction $v$ reads $$d_px_k(v)$$ instead of $$d(x_k)_p(v).$$ Commented Oct 12, 2022 at 16:13