Differentiability of a function between manifolds

Question

Suppose $F:M \to N$ is a function where $M$ and $N$ are submanifolds of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively. According to definition, we know that $F$ is differentiable at $x\in M$ if there are submanifold charts $(W_1,G_1)$ at $x$ and $(W_2,G_2)$ at $F(x)$ such that the map $G_2^{-1} \circ F \circ G_1 : W_1 \to W_2 $ is differentiable at $G_1^{-1}(x) \in W_1$. Does the Differentiability of $F$ depend on the choice of chart? Can we prove that this is not the case. Thank you, in advance, for your response!

Answer 1

It does not depend on the chart. If $(W'_1,H_1)$ is another chart around $x$, then in a neighborhood of $x$ we have that $H_1^{-1}\circ G_1$ is smooth with smooth inverse $G_1^{-1} \circ H_1$. Then $$ G_2^{-1}\circ F\circ H_1 = \left(G_2^{-1}\circ F\circ G_1\right)\circ\left(G_1^{-1}\circ H_1\right) $$

is the composition of smooth functions. Similarly for another chart around $F(x)$.