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An $n$-dimensional Differentiable Manifold is a pair $(X, A)$, where $A$ is a collection of charts $(\varphi_\alpha : V_\alpha \longrightarrow U_\alpha)$ such that the images cover $X$:

$$ X= \displaystyle{ \bigcup_\alpha U_\alpha }$$

where $U_a = \varphi_\alpha(V_\alpha)$ and for every pair of charts $ \varphi_\alpha : V_\alpha \rightarrow U_\alpha$ and $\varphi_\beta : V_\beta \rightarrow U_\beta$, the sets $\varphi^{-1}_\beta(U_\alpha \cap U_\beta)\subset \mathbb{R}^n$ and $\varphi^{-1}_\alpha (U_\alpha \cap U_\beta)\subset \mathbb{R}^n$ are open and the functions

$$\varphi^{-1}_\alpha \circ \varphi_\beta \: : \: \varphi^{-1}_\beta(U_\alpha \cap U_\beta) \rightarrow \varphi^{-1}_\alpha (U_\alpha \cap U_\beta)$$

and

$$\varphi^{-1}_\beta \circ \varphi_\alpha \: : \: \varphi^{-1}_\alpha(U_\alpha \cap U_\beta) \rightarrow \varphi^{-1}_\beta (U_\alpha \cap U_\beta)$$

are differentiable.