In analysis course we encounter commonly the following definition of differentiable function: $f:U \rightarrow \mathbb{R^m}$, where $U \subset \mathbb{R^n}$ is differentiable when
$\exists \ T \in \mathcal{L}(R^m, R^n) $ such that $\forall a \in U$
$f(a + v) - f(a) = T(v) + r(v)$ where $\lim \frac{f(v)}{|v|} = 0$ when $v \rightarrow 0$
However, while studying in Spivak's Differential Geometry he defines differentiable structures for general manifolds, not considered as subsets of some $R^n$. He constructed maximal $C^\infty$-atlases and defined differentiability of a function by coordinated neighborhoods in the manifold.
Do these two definitions agree when the manifold is $R^n$ itself?