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\large{\textbf{Math 4250/6250 Homework \#4}}
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This homework assignment covers our notes on regular surfaces (7-8), tangent planes (9) and the first fundamental form (10). Please pick 5 of the following problems. Remember that undergraduate students should average \textbf{one} challenge problem per assignment, while graduate students should average \textbf{two} challenge problems per assignment. \textbf{Everyone should complete problem \#1 on page 99 as one of your five problems.} This assignment will be due on March 22.
\section{Regular Problems}
\noindent Do Carmo, p 67. \#16, p 88. \#2, \#5, \#10. p 99. \#1.
\section{Challenge Problems}
\noindent Do Carmo, p 99. \#3, \#11, \#12, \#14.
\#14 is a particularly important problem in understanding what the theory of differential geometry is \textit{for}.
\newcommand{\grad}{\nabla}
\newcommand{\del}{\partial}
We might remember that a function $f(x)$ on $\mathbf{R}^n$ has a directional derivative in any direction $v$ given by the limit
\begin{equation*}
Df(v) = \lim_{h \rightarrow 0} \frac{f(x + hv) - f(x)}{h}.
\end{equation*}
We learn in multivariable calculus that this directional derivative is a linear function of $v$, and so that $Df$ is a linear functional on the space of direction vectors $v$. In fact, there is a special vector $\grad f = (\del f/\del x_1, \dots, \del f/\del x_n)$ so that
\begin{equation*}
Df(v) = \left< v, \grad f \right>.
\end{equation*}
If the function $f(x)$ is defined on a curved surface $S$, we still want to be able to understand what it means to differentiate the function. In fact,
the directional derivative at $p$ is a linear function of directions in the tangent plane $T_p S$. This linear functional is now written as
\begin{equation*}
Df(v) = I_p(v, \grad f).
\end{equation*}
But what is the formula for $\grad f$? It is important to know it. But it is
clearly not as simple as it used to be for functions defined on $\mathrm{R}^n$. Finding a formula for $\grad f$ will require us to understand the first fundamental form in some detail, using the theory we've developed. This leads us to an answer to our question ``What is differential geometry for?'':
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\textbf{Differential geometry tells you how to do calculus on a curved surface.}
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