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\noindent {\bf Math 401 HW\#1, due Wednesday 9/4/24} \hspace{.5in} \textbf{NAME:} \hspace{4in} \\
%outlier malcom

\textbf{ Do not turn in the \textcolor{blue}{$\star$ in class} problems. But make sure that you finished and understand them!}   

\begin{enumerate}


%\item Multi Warm Up \\
%Reparametrize the below curve by arc length from $(1,1,0)$ in the direction of increasing $t$
%\[
%r(t)= \langle e^t,  e^{-t}, t\sqrt{2}   \rangle.
%\]
%Then compute $\mathbf{T}, \mathbf{N}, \mathbf{B}$, $k(s)$, and $\tau(s)$.
\item[\textcolor{blue}{$\star$}] \textcolor{blue}{\textbf{Multi Warm Up  (in class)}} \\
Let $v(t) \in \R^n $ such that $|v(t)| = c$ for all $t$.  Show $v' \cdot v =0$.  Conclude $T(t) \cdot N(t) = 0$.

 
\item[\textcolor{blue}{$\star$}] \textcolor{blue}{\textbf{Multi Warm Up: Jones Problem 11-1 (in class)}} \\
Suppose $C \subset \R^2$ is a curve described in polar coordinates by an equation $r=g(\theta)$, where $a\leq \theta\leq b$.  Show that the length of $C$ is
\[
\int_a^b \sqrt{g'(\theta)^2 + g(\theta)^2} \ d\theta.
\] 
%Suggestion: Using polar coordinates, we obtain a parametrization of $C$: $x(\theta)=g(\theta)\cos\theta$, \ $y(\theta)=g(\theta)\sin\theta$. \\
Remark: This formula is usually written as $
\displaystyle \int_a^b \sqrt{\left( \frac{dr}{d\theta}^2 + r^2 \right)} \ d\theta $. 

\item \textbf{Exercise 1.12} (the multi-warm up will be advantageous) \\
 Consider the logarithmic spiral $\alpha:\R \to \R^2$ given by
 \[
  \alpha(t) = (a  e^{bt} \cos t, a e^{bt} \sin t)
  \]
with $a>0, \ b<0.$ Compute the arc length function $S:\R \to \R $ where $S(t) = \int_{t_0}^t |\alpha'(\tau)|d\tau$, where $t_0$ corresponds to an arbitrary choice of $t_0 \in \R$. Reparametrize by arclength (your formula will not be pretty).   Describe/sketch the trace of this curve.   
 
\item {{\bf Remembering Linear Algebra}} 
\begin{enumerate}
\item Provide a counterexample to the claim: For every square matrix $B$, $||Bv|| = | \det B | \ ||v||$.
\item Given an orthogonal matrix $A \in \mbox{O}(n)$, show that $A$ preserves vector norms: $||Av|| = ||v||$.  
\item[](Recall that an orthogonal matrix $A$ satisfies $AA^T = A^TA= \mbox{Id}$.)
\end{enumerate}

\item \textbf{Exercise 1.8} \\ 
 Let $\alpha : I \to \R^3$ be a curve and let $M : \R^3 \to \R^3$ be a rigid motion, e.g. $M = Ax + b$, where $A \in \mbox{O}(3)$ and $b \in \R^3$ is a fixed vector. Prove that rigid motions preserve the length of curves, namely $L_b^a(\alpha) = L_b^a(M \circ \alpha). $
 

\item \textbf{Exercise 1.12} \\
Let $\phi:J \to I$ be a diffeomorphism and let $\alpha: I \to \R^3$ be a curve.  Given $[a,b] \subset J$ with $\phi([a,b]) = [c,d]$ prove that $L_a^b(\alpha \circ \phi) = L_c^d(\alpha)$.



\item[*] \textbf{Assignment Reflections} \\
How long did this take you?  How was the difficulty?  Which problems were meaningful?
\vfill
\eject

\end{enumerate}

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