1
$\begingroup$

I recall reading that using operator (Spivak) style derivative notation, we can unify notation for derivative, partial derivative, Jacobian, and I believe more.

If $f: \mathbb R ^n \to \mathbb R ^m$ (and $n$ and $m$ can of course be $1$), we define $Df$ to be $Df: R ^n \to \mathbb R ^m$. This includes the single-variable derivative and the Jacobian as well.

There was more in the reference, but I can't remember it exactly. They used index and subscripts to pluck out the partial derivatives and other things, without introducing any new notation - it was the same usage of index and subscripts across the board, perhaps the same as used for matrices or vectors. The main idea was subscripts and indices were the same for any $n \to m$ function; derivatives were simply such a function, nothing more, nothing less.

I thought that this was from Spivak, and while I can find his notation starting in this direction, I can't find the whole idea.

So: Is there a reference showing unified notation for derivative operators with indexes and subscripts to unify $n \to m$ functions, derivatives, partial derivatives, and Jacobians, treating them all with the same notation?

$\endgroup$
2
  • $\begingroup$ See also Loomis and Sternberg, Munkres (though Munkres only talks in terms of matrices), Dieudonne, Henri Cartan etc. Also, a remark is that $Df$ does not go from $\Bbb{R}^n\to\Bbb{R}^m$. Rather, $Df:\text{dom}(f)\to \mathcal{L}(\Bbb{R}^n,\Bbb{R}^m)$, so $Df$ is a map with the same domain as $f$, and whose values are linear transformations, so at each point $a$ in the domain, $Df_a$ (or $Df(a)$ as Spivak writes, or $df_a$ as Loomis and Sternberg write or any other notation you fancy (see Loomis for a discussion)) is a map $Df_a:\Bbb{R}^n\to\Bbb{R}^m$. $\endgroup$
    – peek-a-boo
    May 1, 2023 at 2:24
  • $\begingroup$ @peek-a-boo Can you be more specific which books by those authors and where in those books this notation is developed? I can gain access to individual books if you point me in the right direction, but do not yet own most of those books that I can just pull them off my shelf and skim. $\endgroup$ May 1, 2023 at 2:32

1 Answer 1

3
$\begingroup$

A small list of references:

  • Loomis and Sternberg Advanced Calculus. This is freely available online on Shlomo Sternberg’s website. Read chapter 3 (particularly 3.6-3.9). They use the notation $df_a$ for what Spivak would write $Df(a)$.
  • Dieudonne, Foundations of Modern Analysis (this is Volume I of his Treatise on Analysis), chapter 8 is about differential calculus (see 8.1 and 8.9,8.10 for derivatives and partial derivatives and Jacobians). He uses the notation $Df(a)$ for the derivative at a point $a$ of a mapping $f:U\subset E\to F$, where $E,F$ are Banach spaces. In the case $E=E_1\times E_2$ is a product of Banach spaces, he writes $D_if(a)=D_if(a_1,a_2)$ for the partial derivative which is a linear map $E_i\to F$. Anyway, just read the appropriate sections to see various special cases.
  • Henri Cartan Differential Calculus. See Chapter 1, section 2 (particularly 2.1 and 2.6). He uses the notation $f’(a)$ for what Spivak writes $Df(a)$, and $\frac{\partial f}{\partial x_i}(a)$ or $f_{x_i}’(a)$ for the partial derivative (a linear mapping $E_i\to F$), and explains how to relate this to simpler cases when $E_i=F=\Bbb{R}$.
$\endgroup$
4
  • $\begingroup$ Thank you. I've begun taking a look at Loomis & Sternberg (although it's quite advanced). Do any of these references show how you can use subscript indexes consistently to express all of the related notions of derivatives in multiple dimensions? $\endgroup$ May 2, 2023 at 0:40
  • $\begingroup$ @SRobertJames what do you mean subscript indexes? Regardless: notation is just something you’ll have to get used to, the only road to which is practice. So, I suggest reading one book, staying consistent, and solving all the problems. You’ll no doubt get more familiar with everything and how the various concepts are inter-related. $\endgroup$
    – peek-a-boo
    May 2, 2023 at 0:45
  • $\begingroup$ By "subscript indexes" I mean that we use $i$ in $v_i$ to represent an entry of vector $v$, we use $i$ in $A_i$ to represent a row of matrix $A$, we use $i$ in $f_i$ to represent the i-th output of $f$, and we use it in $D_i$ to represent the i-th input of $f$. I believe there are sources which unify most (if not all) of these different meanings. $\endgroup$ May 2, 2023 at 0:53
  • $\begingroup$ Loomis has everything you’ll need (that and Spivak were my main references when I was studying this material). $\endgroup$
    – peek-a-boo
    May 2, 2023 at 1:41

You must log in to answer this question.

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