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\myhead{Mat 5620}{Final Exam}{Fall '14}{ }

Work quickly and carefully, following directions closely. Answer all questions completely. \\[1ex]

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\S I. \textsc{Problems}. 

\begin{enumerate}
\item Prove or disprove: If $\vec{r}\!:\![a,b]\to\mathbb{R}^n$ is integrable, then
	\[ \left\|\, \int_a^b \vec{r}(t)\,dt\,\right\| \le \int_a^b \left\|\vec{r}(t)\right\|\,dt	\]
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\item Prove: If $C$ is a smooth rectifiable curve, and $f$ is continuous everywhere, then
	\[	\left| \int_C f\,ds \, \right| \le \operatorname{length}(C)\cdot \max_{\vec{x}\in C} \left|f(\vec{x})\right|	\]
\vfill

\item Let $\mathcal{S}$ be the surface consisting of the portion of the paraboloid $\Omega = x^2 + y^2$ lying below the plane $z = 1$.  Suppose 
	$F(x,y) = -2y\,\mathbf{i} + 2x\,\mathbf{j} + e^z\,\mathbf{k}$. Use Stokes' theorem to calculate the flux $\displaystyle\iint_{\mathcal{S}} \nabla \times F \cdot \vec n\; dS$.
\vfill

\item Let $\psi(x,y) = \begin{cases} \dfrac{y^2-x^2}{(x^2+y^2)^2} & \text{if $(x,y)\ne(0,0)$} \\ \qquad 0 & \text{if $(x,y)=(0,0)$} \end{cases}$. Show that Fubini's Theorem does not hold for $\psi$ by evaluating
	\[	\int_0^1\int_0^1 \psi(x,y) \,dx\,dy \quad \text{and} \quad \int_0^1\int_0^1 \psi(x,y) \,dy\,dx.	\]
	Explain why Fubini's Theorem doesn't hold for $\psi$ on the rectangle $[0,1]\times[0,1]$.
\vfill

\item Prove: Let $E$ be a subset of $N$ where $\mu(N)=0$. Then $E\in\mathfrak{M}$ and $\mu(E)=0$.
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\item Set $f_n(x)=\dfrac{x^n}{1+x^{2n}}$. Apply Lebesgue's \emph{Dominated Convergence Theorem}, if possible, to determine
	\[	\lim_{n\to\infty} \int_0^\infty f_n \,d\mu	\]
	showing how the theorem applies or fails to apply.
\vfill

\item Prove: Let $f:[0,1]\to\mathbb{R}$ be a non-negative measurable function and $\displaystyle\int_{[0,1]} f\,d\mu=0$. Then $f=0$ almost everywhere.
\vfill

\item Investigate modes of convergence for the following sequences.
	\begin{enumerate}
	\item $g_n:[0,1]\to\mathbb{R}$ given by $g_n(x) = e^n\cdot\chi_{[0,1/n]}(x)$
	\item $h_n:\mathbb{R}\to\mathbb{R}$ given by $h_n(x) = n^{-1}\cdot\chi_{[0,e^n]}(x)$
	\item $k_n:[0,\infty)\to\mathbb{R}$ given by $k_n(x) = \chi_{[n, n+1/n]}(x)$
	\end{enumerate}

\vfill

\item Use \emph{parametric integration} to determine the value of the integral $\displaystyle \int_0^1 \frac{x^{\tau}-1}{\ln(x)}\,dx$ for $\tau>-1$. \\
	{\footnotesize (The ``elementary calculus approach'' substitutes $x=e^{-z}$ for $z=0..\infty$).}
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\end{enumerate}

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