Let $M = \{f(x,y,z) = 0\mid (x,y,z)\in D \}$ be a manifold ($f$ is sufficiently smooth). How I can find border of this manifold analytically? In other words, I want to find parametrization of surface $S = S(u, v)$, where $S$ is a border of $M$.
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Related: http://math.stackexchange.com/questions/2031257/show-that-mathbbbn-is-a-smooth-manifold-with-its-boundary-diffeomorphic-t. – Martín-Blas Pérez Pinilla Nov 30 '16 at 08:54
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If $f$ is sufficiently smooth (and $\nabla f\ne 0$), the level set $M$ will be a manifold without border (or with empty border if you want). You can get a manifold with border "cutting" $M$ with another smooth function $g$ s.t. $0$ is a regular value of $g$: $$N = \{(x,y,z)\mid f(x,y,z) = 0, g(x,y,z)\ge 0\}.$$ The border will be the set of points $p\in M$ s.t. $g(p) = 0$. A parametrization of $\partial N$?. The implicit function theorem says that exists.
Martín-Blas Pérez Pinilla
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His "border" is coming from the boundary of the parametrizing domain $D$, it seems to me. So we would need to parametrize $\partial D$. – Ted Shifrin Nov 30 '16 at 21:40