Smooth map

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Definition

A map $f:M\rightarrow N$ between two smooth manifolds $(M,\mathcal{A})$ and $(N,\mathcal{B})$ (of not necessarily the same dimension) is said to be smooth if:

• $$\forall p\in M\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}$$ such that $$F(U)\subseteq V\wedge[\psi\circ F\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)]$$ is smooth

See also

• Differential of a smooth map
• Smooth function
• Smooth manifold