Tsinghua University 2016 direct examination test questions

Special paper for math test papers

April 2016

i) set $d$ to a region on the $\mathbb{r}^n$ $f:d\to \mathbb{r}^n$ is a continuous micro-mapping. The inverse mapping theorem (including conditions and conclusions) about map $f$ is described .

II) Try to use inverse mapping theorem to prove that there is no continuous micro-injection from $\mathbb{r}^n$ to $\mathbb{r}^1$.

second, the vector field on a given $\mathbb{r}^3\backslash\{0\}$

\[\overrightarrow v = \left ({\frac{x}{{{{\left ({x^2} + 2{y^2} + 3{z^2}} \right)}^{\frac{3}{2}}}}},\frac{y}{{{{\le ft ({{x^2} + 2{y^2} + 3{z^2}} \right)}^{\frac{3}{2}}}}},\frac{z}{{{{\left ({{x^2} + 2{y^2} + 3{z^2}} \right)}^{\frac{3}{2} }}}}} (\right). \]

The $\overrightarrow n$ is the unit outer normal vector field of the unit spherical $s^2$ in $\mathbb{r}^3$. Try to find the integral

\[\int_{s^2}\overrightarrow V\cdot \overrightarrow n d\sigma.\]

third, set the function defined on the $\mathbb{r}$ cycle is $2\pi$ $f$ on the interval $ (-\pi,\pi]$ value is $f (x) =x$.

i) try to give its Fourier series, find out the sum function of Fourier series, and indicate whether this series is uniformly convergent on $\mathbb R.

II) using the Fourier series and Parseval equation to calculate the sum of series $\sum_{n\geq1}\frac1{n^2}$.

$f (x) $ is resolved on the unit disk $|z|<1$, satisfies $|f (z) |<1$, and $f (\alpha) =0$, of which $|\alpha|<1$.

1. Try to prove that when $|z|<1$ is established \[\left| {F\left (z \right)} \right| \le \left| {\frac{{z-\alpha}}{{1-\overline \alpha Z}}} \right|. \]

2. Give the necessary and sufficient conditions for the formation of an equal sign in the above inequalities .

v. Given $a\in m_n (\MATHBB C) $. Make $f (x) $ for its characteristic polynomial, $g (x) \in \mathbb C [x]$ is a $f sub-polynomial that divides the $n-1$ (x) $. Ask for a possible rank of $g (a) $ and explain the reason.

The $v$ is a $n$ dimensional linear space on the complex field, and the $\sigma$ is a power unitary transformation on $v$ (i.e., there is a positive integer $k$ makes $\sigma^k=1_v$, $1_v$ is the identity transform on $v$). $w$ for $v$ Invariant subspace. Proves that there is $\sigma-$ invariant subspace in $v$ $w ' $ makes $v=w\oplus W ' $.

Transferred from: http://www.math.org.cn/forum.php?mod=viewthread&tid=36021

Tsinghua University 2016 direct examination test questions