Confused about "total derivative"

Question

I'm confused about "total derivative", and looking around, I'm not the only one.

I have seen this https://math.stackexchange.com/a/174272/290307 .

This answer is obviously flawed by not distinguishing "function" and "expression",

The function has fixed positions for $x$ and $y$, for $f:\;\;\mathbb R\times \mathbb R\,\mapsto \mathbb R$, whereas the expression, something like $2x+3y$, without the whereabouts of $x$ and $y$, has not.

I am looking for a solid definition of the total derivative, not what it is used like colloquially.

I don't understand why the difference of function vs. expression comes up. Seems to have nothing to do with it. — Randall, Oct 19 '20 at 17:30
Anyway, the total derivative is a linear transformation (or equivalently, a matrix). Any good answer would have to use those tools. Are you familiar with them? — Randall, Oct 19 '20 at 17:31
In what generality do you want to have the definition? In euclidean space or general Banach spaces? — Severin Schraven, Oct 19 '20 at 17:31
@Randall I have a degree in math, but I'm decades out of practice. — Gyro Gearloose, Oct 19 '20 at 17:34
The clearest definition/discussion I've seen is in Baby Rudin. — Randall, Oct 19 '20 at 17:36
@SeverinSchraven finite real vector spaces would be good enough for a start. More is always welcome, but I can't give a guaranty that I would have understanding of it. — Gyro Gearloose, Oct 19 '20 at 17:40
@Randall "The clearest definition/discussion I've seen is in Baby Rudin." Could you spare a link to it? My google search (newly requiring me to allow data collection totally monitoring me, and you) did not come up with relevant results. — Gyro Gearloose, Oct 19 '20 at 18:37

score 1 · Accepted Answer · answered Oct 19 '20 at 18:40

Let $U\subseteq \mathbb{R}^n$ be an open, nonempty set, $f: U\rightarrow \mathbb{R}^m$ be a function and $x_0\in U$ some point. We say that $f$ is (Frechet-)differentiable in $x_0$ if there exists a linear function $L_x: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $$ \lim_{h\rightarrow 0} \frac{\Vert f(x_0+h)-f(x_0)-L_x(h) \Vert_{\mathbb{R}^m}}{\Vert h \Vert_{\mathbb{R}^n}} =0.$$ In this case we call $L_x$ the total derivative of $f$ at the point $x_0$. Usually we write $Df(x_0)$ instead of $L_x$.

So what is this total derivative? It is the best "linear approximation". I.e. if you want to fit some linear function at $f$ in $x_0$, then you want to pick the total derivative. How is this related to the "usual" derivative we see in college? If we have $g:\mathbb{R} \rightarrow \mathbb{R}$ which is differentiable in $x_0$, then $g'(x_0)$ is the slope of the tangent at $g$ in $x_0$. This line is the best fit you can have at $g$ in $x_0$. In one dimension the line is uniquely determined by its slope (and the fact that it has to go through $(x_0,g(x_0))$. Hence, there is the following correspondence between derivative and total derivative in one dimension: $g'(x_0)= Dg(x_0)[1]$. Or in other words $$ Dg(x_0):\mathbb{R}\rightarrow \mathbb{R}, Dg(x_0)[h] = g'(x_0) \cdot h. $$

Confused about "total derivative"

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