Derivative operator and mapping to a homomorphism I don't understand the expression
$$\begin{align}D(\cdot)_a:\mathcal{D}_{a,U,\Bbb{R}^m}&\to\text{Hom}(\Bbb{R}^n,\Bbb{R}^m)\\
f&\mapsto Df_a
\end{align}$$
intended to denote the derivative-at-point-a operator, where $U\subset\Bbb{R}^n$ and $a \in U.$ This appears here.
If it is at all possible to focus on the operator that maps a function to its derivative, leaving aside the rest of the moving parts, my only approximation to what this means is that for real polynomials
$$\begin{align}D: \mathbb R[x]&\to\mathbb R[x]\\
P&\mapsto P'
\end{align}$$
forms a group homomorphism with addition.
However, this is not the case for general real-valued differentiable functions as in here.
So can someone explain in simple terms what the first expression denotes?
 A: What you call approximation of what happens is actually something different. The map $$D:\mathbb R[x] \to \mathbb R[x]\\ P \mapsto P'$$
Is a endomorphism of vectorspaces and doesn't really rely on the analytic concept of a dreivative, just on the formal dericative of polynomials that $(x^n)' = D(x^n) = nx^{n-1}$. Here you map a polynomial to its formal derivative, which is something global.
Then on the other hand you have the map $D(.)_a$ from totally differentiable functions on an open neighborhood $U\subset \mathbb R^n$ to $\mathbb R^m$. This operator maps each such function $f$ to its total derivative at the point $a$, this total derivative is itself a linear map from $\mathbb R^n \to \mathbb R^m$ so it lies in $Hom(\mathbb R^n, \mathbb R^m)$. This operator $D(.)_a$ is itself linear, but here we are interested in the best approximation of $f$ by a linear function, the total derivative. This is something we are doing locally, as different functions can ave the same derivative at some point. In case of $n=m=1$ this is the approximation of our function by its tangent. (To be completely accurate to get the linear approximation we still need to "move" our total derivative to the point $a$, as until now it would still "sit" at the origin)
In general the linear approximation looks like this:
Let $f$ be like above than we call $f$ totally differentiable in $a$ if there exists a linear map $D_a:\mathbb R^n\to \mathbb R^m$ and a in $a$ continuous function $r:U\subset\mathbb R^n\to \mathbb R^m$ such that:
$$f(x) = f(p) +D_a(x-p)+|x-p|r(x)$$
Then $D_a(f) = D_a$, this is a equivalent definition to others and it shows how we approximate $f$ by $D_a(f)$, this eventually leads to the multidimensional Taylor-polynomial.
So to conclude on one side we have the global homomorphism $D$ on polynomials that maps each polynomial to its derivative-polynomial and we have the local linear operator $D_a$ that maps multidimensional real-valued functions to their derivative in a point $a$.
Remark: You can have something similar to mapping a function to its derivative function as you have it in the one-dimensional case, you can map $f$ to $Df(x)$, where $Df(x) = D_x(f)$. This function also takes values in $U$, but maps to $Hom(\mathbb R^n, \mathbb R^m)\cong \mathbb R^{nm}$, by choosing a basis, this is also referred to as the total derivative or also its Jacobian matrix. It is continuous if $f$ is continuously differentiable.
