A function $f: U\subseteq\Bbb R\to \Bbb R$ is differentiable at a point $p\in U$ if $f(p+h) = A + Bh + o(h)$ for $p+h$ in a neighborhood of $p$. It can be shown that $A$ and $B$ are uniquely determined and $A=f(p)$ and $B=f'(p)$.
This is a convenient way to define differentiability and prove facts about the derivative. Can we generalize the little o definition of differentiability/ the derivative for use with functions $g:U\subseteq \Bbb R^m \to \Bbb R^n$?
The problem I can see encountering is this. The way I was shown that this is equivalent to the usual definition of differentiability of scalar functions is by setting $\frac{f(p+h)-f(p)-f'(p)h}{h}$ equal to a function $r(h)$ such that $\lim_{h\to 0} r(h) = 0$. Then just multiply by $h$ and add $f(p)+f'(p)h$ to get the asymptotic equation.
But in higher dimensions the differentiability condition has those (annoying) norms $$\lim_{h\to 0} \frac{\|f(p+h)-f(p)-Df(h)\|_{\Bbb R^n}}{\|h\|_{\Bbb R^m}}=0$$ Specifically I don't know what to do about the norm in the numerator. So I don't think I'd be able to use the same process.
Yes. $f : U \subseteq \mathbb{R}^n \to \mathbb{R}^m$ is differentiable at $x \in U$ if there exists $J \in \mathbb{R}^{m \times n}$ such that
$$f(y)=f(x)+J(y-x)+o(\| y - x \|)$$
as $y \to x$. $J$ is called the Jacobian matrix of $f$ at $x$, sometimes denoted by $Df(x)$. One can prove that $J$ is unique and that the choice of the norm used in the above does not matter.
The situation is very similar for maps between infinite dimensional Banach spaces, but there the details become more complicated.