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\title{Math 147A: Homework 4}
\date{}


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\begin{document}
\maketitle
\textbf{Due Thursday, May 12.}

Read Chapter 4
\begin{enumerate}
\item (4.2.1) Show that, if $f(x,y)$ is a smooth function, its graph
\begin{equation*}
\{(x,y,z) \in \mathbb{R}^3 \ | \ z = f(x,y)\}
\end{equation*}
is a smooth surface with atlas consisting of the single regular surface patch
\begin{equation*}
\sigma(u,v) = (u,v,f(u,v)).
\end{equation*}
\item (4.2.3) Which of the following are regular surface patches:
\begin{enumerate}
\item $\sigma(u,v) = (u,v,uv)$
\item $\sigma(u,v) = (u,v^2,v^3)$
\item $\sigma(u,v) = (u+u^2,v,v^2)$
\end{enumerate}
\item (4.2.7) Let $\gamma$ be a unit-speed curve in $\mathbb{R}^3$ with nowhere vanishing curvature.  The tube of radius $a>0$ around $\gamma$ is the surface parametrized by
\begin{equation*}
\sigma(s,\theta) = \gamma(s) +a(\mathbf{n}(s)\cos(\theta)+\mathbf{b}(s)\sin(\theta)),
\end{equation*}
where $\mathbf{n}$ is the principal normal of $\gamma$ and $\mathbf{b}$ is its binormal.  Give a geometrical description of this surface.  Prove that $\sigma$ is regular if the curvature $\kappa$ of $\gamma$ is less than $a^{-1}$ everywhere.
\item (4.4.1) Find the equation of the tangent plane of each of the following surface patches at the indicated points:
\begin{enumerate}
\item $\sigma(u,v) = (u,v,u^2-v^2), \quad (1,1,0)$
\item $\sigma(r,\theta) = (r\cosh\theta,r\sinh\theta,r^2), \quad (1,0,1)$
\end{enumerate}
\item (4.4.3) Let $S$ be a surface, let $p\in S$ and let $F:\mathbb{R}^3 \to \mathbb{R}$ be a smooth function.  Let $\nabla_S F$ be the perpendicular projection of the gradient $\nabla F$ of $F$ onto $T_pS$.  Show that, if $\gamma$ is any curve on $S$ passing through $p$ when $t=t_0$, say
\begin{equation*}
(\nabla_S F)\cdot\dot{\gamma}(t_0) = \frac{d}{dt}\biggr|_{t=t_0} F(\gamma(t)).
\end{equation*}
Deduce that $\nabla_S F=0$ if the restriction of $F$ to $S$ has a local maximum or a local minimum at $p$.
\end{enumerate}

\end{document}