Interpretations of the derivative

Question

In class we were told to think of the derivative of a function in higher dimensions as a linear transformation between the tangent spaces of $\mathbb{R^n}$, that is $$Df_p:T_p\mathbb{R^n} \to T_p\mathbb{R^m}.$$

But we also introduced the concept of the dual space to help explain the derivative: the way I understand it, the derivative is an element of the cotangent space, so $$Df_p:(T_p\mathbb{R^n})^* \to \mathbb{R}.$$

Also, we were told that we could interpret the derivative as $$Df_p: (T_p\mathbb{R^n})^* \times T_{f(p)}\mathbb{R^m} \to \mathbb{R}.$$ These definitions seem contradictory to me. I'm having trouble interpreting them, and how they could relate to each other.

Answer 1

$\newcommand{\R}{\mathbb R}$

1. Derivative of $f:\R^n\to\R^m$ at $p$ is a linear map $Df_p:T_p\R^n \to T_{f(p)} \R^m$

2. Derivative of $f:\R^n\to\R$ at $p$ is a linear map (a functional) $Df_p:T_p\R^n \to T_{f(p)} \R\simeq\R$, and indeed is an element of the cotangent space, $Df_p\in T^*_p\R^n$.

   2(a). Duality pairing is the bilinear map $V^* \times V \ni (\alpha,u)\mapsto \alpha(u) \in \R$. In our case $V$ is $T_p\R^n$, and we can take $\alpha=Df_p$, but we cannot reuse the symbol $Df_p$ for the bilinear map itself!