= Smooth embedded surface
A smooth embedded surface in $\mathbb R^3$ is a two-dimensional <smooth manifold> locally represented by an injective regular <parametrized surface> $\sigma:U\subset\mathbb R^2\to\mathbb R^3$, with $\sigma_u\times\sigma_v\ne0$. The <first fundamental form> measures lengths and angles, and a choice of <unit normal> determines the <second fundamental form>, which measures bending. A surface with boundary additionally uses local half-plane charts along its boundary.
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