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Which of the following are regular surface patches? Let $u,v\in \mathbb{R}$.

$(i)$ $σ(u, v)=(u, v, uv)$

$(ii)$ $σ(u, v)=(u, v^2, v^3)$

$(iii)$ $σ(u, v)=(u + u^2, v, v^2)$

I'm not sure how to show this. I think if $σ$ is smooth and $σ_u \timesσ_v$ is nowhere zero, then $σ$ is regular. Any hints are greatly appreciated, thanks.

Gino
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1 Answers1

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Proposition. Let $p \in S$ be a point of a regular surface S and let $X$:$U \subset \mathbb R^2 \to R^3$ ($U$ an open set of $\mathbb R^2$) be a map with $p\in S$ such that: (S1) $X$ is differentiable; (S3) For each $q \in U$ $X_u$$\times$ $X_v$ is nowhere zero and assume that $X$ is one-to-one. Then $ X^{-1} $ is continuous.
Basically this means that $X$ is a regular paraetrization of S. So you only need to prove, for each case, (S1),(S3) and that $\sigma$ is one-to-one.
I think that is all.

Luis Victoria
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  • You got the logic wrong: The condition (S1), (S3) are necessary but not sufficient. The standard example (for curves) is the "figure 8" (see here) which easily generalizes to maps of surfaces into $R^3$. – Moishe Kohan Mar 12 '22 at 09:37