Manifold
n -Sphere as a patchwork

The n -sphere S is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An n -sphere S can be constructed by gluing together two copies of R n {\displaystyle \mathbb {R} ^{n}}. The transition map between them is inversion in a sphere, defined as R n ∖ { 0 } → R n ∖ { 0 }: x ↦ x / ‖ x ‖ 2. {\displaystyle \mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.}

This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function, this atlas defines a smooth manifold.In the case n = 1, the example simplifies to the circle example given earlier.
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