I'm trying to understand some questions about the chain rule of multivariable and how it relates with one variable functions.
Ok, so, imagine that I have two functions $f: U \subset E \rightarrow \mathbb{R}$ and $\alpha: (-\epsilon, \epsilon) \subset \mathbb{R} \rightarrow E$, where $E$ is a normed vector space.
Then, by the chain rule, the derivative of $f \circ \alpha$ in some point $a \in U$ is:
$(f \circ \alpha)'(a) = Df_{\alpha(a)} \cdot D\alpha_a$.
The point that I don't get is that $(f \circ \alpha)'(a)$ is a number, but $Df_{\alpha(a)} \cdot D\alpha_a$. is a linear application. So, I'm a little bit confused, can the number $(f \circ \alpha)'(a)$ be interpreted in some way by a linear application?
Thanks!
The derivative of a function $f:\mathbb R\to\mathbb R$ at $x\in\mathbb R$ is a number $c\in\mathbb R$ characterized by the fact that $$f(x+h)=f(x)+ch+o(h^2).$$ The term $ch$ is essentially a linear map $h\mapsto ch$, which is entirely characterized by the number $c$, which is the derivative of $f$. That's how the derivative of a 1d function should be interpreted as a linear map. More conceptually, the tangent function $t(x+h)=f(x)+f'(x)h$ is a linear map (affine linear, to be exact) as a function of $h$, determined by the derivative. The actually linear part $f'(x)h$ is the linear map we're talking about.
For higher dimensions, this works exactly the same: If $\mathrm Df_x$ is the derivative of $f$ in $x$, then $$f(x+h)=f(x)+\mathrm Df_x(h)+o(\vert h\vert^2).$$ Just like in the 1d case.