Unified notation for derivatives, partial derivatives, Jacobians, and more

Question

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?

Answer 1

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 $E_i=F=\Bbb{R}$.