\documentclass{math412}

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\usepackage{times,txfonts}

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\doclabel{Math 412: Demo Homework}
\docauthor{Your name here.}
\docdate{March 14, 2013}

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\input{math412extras.tex}


\newcommand{\vv}{\mathbf{v}}
\begin{document}

\begin{exercise}{Oprea 2.2.8}
Suppose $M$ is a surface which is the level set of a function $g$, $g(x,y,z)=c$.
Show that $\nabla g(p)$ is a normal vector for $M$ at every $p\in M$.
\end{exercise}
\solution
Let $\vv \in T_p M$.  Let $\alpha$ be a curve in $M$ with $\alpha(0)=p$ and $\alpha'(0)=v$.
Note that since $\alpha$ is a curve in $M$, and since $g$ is constant on $M$, $g(\alpha(t))$
is constant for all $t$.  Hence
$$
\vv[g] = \left.\frac{d}{dt}\right|_{t=0} g(\alpha(t)) = 0.
$$
On the other hand, on the previous homework we showed that
$$
\frac{d}{dt}g(\alpha(t)) = \nabla g(\alpha(t))\cdot \alpha'(t).
$$
Evaluating this at $t=0$, using the facts that $\alpha(0)=p$ and $\alpha'(0)=\vv$ we have
$$
0 = \vv[g] = \nabla g(p) \cdot v.
$$
Since $v\in T_p(M)$ is arbitrary, $\nabla g(p)$ is normal to $M$ at $p$.

\end{document}