6
$\begingroup$

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

$\endgroup$
1
  • 1
    $\begingroup$ Re. your second question, $\frac{\partial f(p)}{\partial p}$ is definitely not correct because $p$ is fixed. Related $\endgroup$
    – user170231
    Commented Oct 11, 2022 at 17:27

1 Answer 1

Reset to default
8
$\begingroup$

$\frac{\partial f}{\partial x}$ 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 would also avoid calling the total derivative the Jacobian; they are conceptually not the same thing. The total derivative is a linear transformation and the Jacobian is a matrix describing that linear transformation with respect to a suitable choice of bases.

$\endgroup$
6
  • $\begingroup$ 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 $\mathrm{d}f(p)(v)$. $\endgroup$
    – Didier
    Commented Oct 11, 2022 at 16:52
  • $\begingroup$ 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, $y=f(x)$? $\endgroup$
    – J Peterson
    Commented Oct 11, 2022 at 17:13
  • 1
    $\begingroup$ @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 $\left( \frac{\partial f_j}{\partial x_i}\right)_{1\leqslant i \leqslant n, 1\leqslant j \leqslant m}$. $\endgroup$
    – Didier
    Commented Oct 11, 2022 at 17:24
  • 2
    $\begingroup$ @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. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 11, 2022 at 17:29
  • 1
    $\begingroup$ 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 $$d(x_k)_p(v).$$ $\endgroup$
    – Michael Hoppe
    Commented Oct 12, 2022 at 16:13

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .