Smooth Manifold
Overview
Is a special type of topological manifold where the charts have compatible versions of calculus.
Definition
A smooth manifold is a topological manifold such that for two charts
\phi_\beta:U_\beta \to \mathbb{R}^n$$ (defined on open sets $U_\alpha, U_\beta \subseteq M$) $$\phi_\beta \circ \phi_\alpha^{-1}\big|_{\phi_\alpha(U_\alpha \cap U_\beta)}: \phi_\alpha(U_\alpha \cap U_\beta) \subset \mathbb{R}^n \longrightarrow \phi(U_\alpha \cap U_\beta)\subset \mathbb{R}^n$$ is smooth (i.e. $\phi_\beta \circ \phi_\alpha^{-1} \in C^\infty(\mathbb{R}^n)$ with the necessary restrictions). Thus, where the domains of the charts overlap, we know "calculus works" and the smooth structure can vary between charts.