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Section 7 Wrap Up Questions

  1. For a given surface \(S\text{,}\) how many different parameterizations of \(S\) are there?
  2. How many different ways are there to put an orientation on a surface \(S\text{?}\)
  3. Given a parameterization of a surface, how would you go about finding a parameterization with the opposite orientation?
  4. Explain why if the graph of a function \(z=f(x,y)\) is parameterized as
    \begin{equation*} \vec{r}(u,v)= u\vec{i} +v\vec{j} +f(u,v)\vec{k} \end{equation*}
    then, provided \(f(x,y)\) is differentiable, the normal vector \(\vec{N}\) is never \(\vec{0}\) and always points in an upward direction.
  5. For a surface \(S\) parameterized by
    \begin{equation*} \vec{r}(u,v) =f(v) \cos{(i)} \vec{i} +f(v)\sin{(u)} \vec{j} +v\vec{k}, \end{equation*}
    where \(f(v)\) is a function of height \(v\text{,}\) can you determine conditions for when \(\vec{r}(u,v)=\vec{N}\text{?}\) Think geometrically about this question.
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