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\myhead{Mat 5620}{Homework I}{Spring '14}{}

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\S I. There are 2 proofs at 5 points each.
\begin{proposition}
Suppose $\vec{r}(t)$ is integrable on $[a,b]$. Then $ \left\| \vec{r}(t) \right\|$ is integrable on $[a,b]$ and
\[	\left\| \int_a^b \vec{r}(t)\,dt \right\| \le\int_a^b  \left\| \vec{r}(t) \right\| \,dt.	\]
\end{proposition}
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Let $\vec{r}(t)=\langle f(t),g(t)\rangle$ for $t\in[a,b]$. Define the \emph{curvature} $\kappa$ of $\vec{r}$ at $t$ to be
\[	\kappa = \left| \frac{d\phi}{ds}\right|	\]
where $\phi=\arctan(dy/dx)$ \emph{(the angle between the tangent vector and the $x$-axis)} and $ds=\sqrt{dx^2+dy^2}$. Use 
\[	\left| \frac{d\phi}{ds}\right| = \left| \frac{d\phi/dt}{ds/dt}\right|	\]
to prove (remembering that for parametric functions in $\mathbb{R}^2$, $dy/dx = (dy/dt)/(dx/dt)$)
\begin{proposition} For $\vec{r}(t)=\langle f(t),g(t)\rangle$ twice differentiable for $t\in[a,b]$, 
\[	\kappa =  \frac{\left| \frac{dx}{dt}\,\frac{d^2y}{dt^2} - \frac{d^2x}{dt^2}\,\frac{dy}{dt}\right| }{\phantom{^{3/2}}\left[ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 \right]^{3/2}}	\]
\end{proposition}

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