Taylors theorem from $R^m$ to $R^n$

I am trying to understand the first order taylor approximation for functions of several variables from $R^m$ to $R^n$. But, I can’t find a single source online!

$$f(x) = f(a) + Df(c)(x-a)$$Here $Df(c)$ is the Jacobian of $f$ at $c$. $c$ is any point in the “line segment” joining $x$ and $a$. This is just the Mean Value Theorem

My question is:

• Now if I change $Df(c)$ to be approximated by $Df(a)$, what would be the reminder term? I know what happens for $R^m \rightarrow R$ case but what about $R^m \rightarrow R^n$.

Lastly, is there a source/book that deals with taylor's theorem in $R^m \rightarrow R^n$?