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\vspace{-1in}
\begin{center}
 \large\bf \m Math 366,  Final Test 
$\left.\hspace{4em}\oval(80, 40)\right/ 130$
 \end{center}
\begin{flushright}
May 8,  1997\\
Instructor: Bei Hu
\end{flushright}

\hspace{.5\textwidth} Name:\hrulefill

\vspace*{2ex}

1. (30 points).  

(a). State PRECISELY the definition of an k-form.
\vspace{2in}

(b). State PRECISELY the Stone-Weierstrass theorem in a rectangle in 
$R^2$.
\vspace{2.5in}

(c). State PRECISELY the Stocks theorem.
\vspace{2in}

\newpage

(d). State PRECISELY the theorem of partition of unity.
\vspace{2in}
 
(e). State PRECISELY a necessary and sufficient condition
for the existence of a Riemann-Stieltjes integral. 
\vspace{2in}
  
(f). State PRECISELY the Ascoli Arzel\'a theorem.
\newpage

2. (12 points). 
 In $R^3$, we use $(x,y,z)$ variables.
Let $\la =  y dx\wedge dz  $.
Let $\Phi $ be a 2-surface $x = u_1, y=u_2, z= 1-u_1-u_2$,
$u_1\ge 0$, $u_2\ge 0$, $u_1 + u_2 \le 1$.
Find
$$ \int_\Phi \la. $$

\newpage


3. (12 points).
Is it true that every uniformly convergent sequence of bounded functions is 
uniformly bounded?  Prove or give a counter example.

\newpage

4.  (12 points)

Suppose that $f$ is Riemann integrable on $[0,A]$ for any $A>0$ and
$f(x)\to 1$ as $x\to +\infty$. Prove that
$$
 \lim_{t\to 0+} t\int_0^\infty e^{-tx} f(x) dx = 1 .
$$
\newpage
5. (12 points)


Consider
\begin{eqnarray*}
&& f_1(x_1,x_2, y_1,y_2,y_3) =  e^{x_1} + x_2 y_2 - 9y_1 y_3  = 0\\
&& f_2(x_1,x_2, y_1,y_2,y_3) = 5x_2  g(x_1) + g(x_1+2)  + 2 y_2 + 8y_3 =0,
\end{eqnarray*}
where $g(x_1)$ is continuously differentiable. We want to solve 
$(x_1,x_2)$ in terms of $(y_1, y_2, y_3)$ near $(0,0,0,0,0)$.

 Find the condition(s) on $g(x_1)$ so that the implicit function theorem
 can be used near $(0,0,0,0,0)$.




\newpage

6. (12 points). Suppose that $\la = -y dx + x dx $.
Suppose that $\O$ is a simply connected domain with smooth
boundary. What can you conclude from the Stokes theorem?

Explain your result.

\newpage


7. (12 points).
Consider the family of continuous functions
$f_n(x,y) = \dis \frac{ x^2 + y^2 + 5xy }{x^2 + y^2 + (1/n)^2}$
on the disk $B_1 = \{(x,y)|x^2+y^2 \le 1\}$. 

It is clear that $f_n(x,y)$ converges to some $f(x,y)$ pointwise
on $B_1$.

(a) Show that $f(x,y)$ is not continuous.

(b) Is the convergence $f_n(x,y)\to f(x,y)$ uniform on $B_1$?
Justify your result.

\newpage

8. (10 points). 
Let $f(x) = x + e^{-x/2}$ and $X=[0,\infty)$.
We claim that $f(x)$ has a fixed point in $X$, by the contraction
mapping principle.

{\bf Proof.}  It is clear that for $x\in X$,
$ f'(x) = 1 - \dis\frac12 e^{-x}$, and therefore $0<f'(x)<1$.
It follows that, by Mean value theorem
$$
  |f(x)-f(y)| = |f'(\xi)|<1 \for x,y\in X.
$$
Therefore the contraction mapping principle will give a unique
fixed point for $f(x)$.\q

Is this proof correct? Why or why not.


\newpage
9. (10 points).
Suppose that $f(x)\ge 0$ for $x\ge 0$ and is monotonically decreasing.
Prove that $\dis\int_0^\infty f(x)dx$ converges if and only if
$\dis \sum_{n=1}^\infty f(n)$ converges.


\newpage
10. (8 points).
 Suppose that $f(x)$ is continuous and 
$$
 \int_E f(x) dx = 0 
$$
for any open set $E$ in $R^n$. Prove that $f(x)\equiv 0$.

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