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\noindent {\bf Math 222 HW\#7, due Friday 4/2/21} \hspace{.5in} \textbf{NAME:} \hspace{4in} \\
%outlier malcom

\smallskip
\noindent Do 1-5 or 3-6. Prof Jo is happy to give hints on \#6; please ask via a discussion in Canvas.  
\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 Problem 8-3: Jones\\
%For the record, here is the general principle used above.  Prove that for a general real symmetric $n\times n$ matrix $A$ which is positive semi-definite and for $t\in \R^n$, 
%\[
%At \cdot t = 0 \mbox{ if and only if } At=0.
%\]
%(It will be easiest to do each direction separately.)


\item Compute $\displaystyle \int_C (y^2dx - 2x dy)$ where $C$ is the triangle with vertices $(0,0)$, $(1,0)$, and $(1,1)$, oriented counterclockwise (Folland: $-4/3$).  Do so by 
(a) Directly parametrizing $C$;  (b)  Green's theorem.  

\item Compute $\displaystyle \int_C F \cdot d\mathbf{x}$, where $F = (x^2y, x^3y^2)$ and $C$ is the closed curve formed by portions of the line $y=4$ and the parabola $y=x^2$, oriented counterclockwise. Do not use Green's theorem. (Folland: $2(\frac{32}{5} + \frac{1024}{9} - \frac{32}{3})$)

\item Jones Problem 11-1 \\
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 
\[
\int_a^b \sqrt{\left( \frac{dr}{d\theta}^2 + r^2 \right)} \ d\theta
\] 


\item Curves and Surfaces computation (made easier by Jones 11-1 if you convert to polar and use formula) \\
 Consider the logarithmic spiral $\mathbf{r}:\R \to \R^2$ given by
 \[
  \mathbf{r}(t) = (a  e^{bt} \cos t, a e^{bt} \sin t)
  \]
with $a>0, \ b<0.$ Compute the arc length function $S:\R \to \R $ where 
\[
S(t) = \int_{t_0}^t |\mathbf{r}'(\tau)|d\tau
\]
for $t_0$ corresponding to an arbitrary choice of $t_0 \in \R$.  Describe/sketch the curve.  
 
 
\item Jones 12.E, 12-7 \\ 
Define a 1-form $\alpha$ on the punctured plane $\R^2 \setminus \{ 0 \}$ by $
\alpha = \left( \dfrac{-y}{x^2 + y^2} \right) dx +  \left( \dfrac{x}{x^2 + y^2} \right) dy. $
\vspace{-.125in}
\begin{enumerate}
\item
 Calculate $\int_C \alpha$ for any circle $C$ of radius $R$ around the origin. 

\item Prove that in the half plane $\{ x >0 \}, \ \alpha$ is the differential of a function.  
\end{enumerate} 

\item Curves and Surfaces short proof  \\
 Let $\mathbf{r} : I \to \R^3$  be a $C^1$ curve and $[a,b]\subset I.$ Prove that 
\[
|\mathbf{r}(a) - \mathbf{r}(b)| \leq \int_a^b |\mathbf{r}'(t)|dt,
\]
e.g. straight lines are the shortest curves joining two given points.  Hooray!  \\
Suggestion: First, prove that for any constant vector $u \in \R^3$, that 
\[
\left \langle \int_a^b \mathbf{r}'(t)dt, u \right \rangle \leq |u| \int_a^b |\mathbf{r}'(t)|dt.
\]
Then use the Fundamental Theorem of Calculus after plugging in $u =  \int_a^b \mathbf{r}'(t)dt$ to the preceding inequality that you proved.  Be careful with your inequalities and absolute values!  



\item[*] Assignment Reflections\\
 How difficult was this assignment?  How many hours did you spend on it?  Which problems did you find to provide a worthwhile learning experience?  Should I be assigning a similar number of problems, fewer problems, or more problems in the future?  Is there a good mix of theory and computations?

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\end{enumerate}

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