user278130
6
A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n.
Is there a citeable reference for this result?
For the sake of being definite, let's say that
“citeable” means indexed by Mathematical Reviews or Zentralblatt,
or available on arXiv.
(The answer https://mathoverf...
Ad question 1): Yes, all open star-shaped subsets of $\mathbb{R}^n$ are diffeomorphic to $\mathbb{R}^n$.
This is surprisingly little-known and there is a proof due to Stefan Born.
You can find this (fairly complicated) proof in Dirk Ferus's course notes
http://www.math.tu-berlin.de/~ferus...
If $F$ is a field and $a \neq 0$ is a zero of $f(x) = a_{0} + a_{1}x + · · · + a_{n}x
^n$ in $F[x]$, Then $1/a$ is a zero of $a_{n} + a_{n−1}x + · · · + a_{0}x^{n}$.
I thought of this -
As $a$ is a zero of $f(x)$ so $f(a) = a_{0} + a_{1}a + · · · + a_{n}a
^n = 0$
Now as $a \in F$ so inverse of ...
For the integration I tried different substitution but they all are useless .
And I am not getting any other method .
Can anybody provide me a hint
Determine $$\iint_S \textbf{A} \cdot d\textbf{S}$$
where $$\textbf{A}(x,y,z) = (xz^2,x^2y,xyz)$$ and S is the sphere $(x-1)^2+(y-2)^2+(z+3)^2=4$
Attempted solution
Let's use Gauss' theorem. We obtain $\nabla \cdot \textbf{A} = x^2+z^2+xy$. A parametrization of the sphere is given by:
$$\Phi(r,\...
Let $z_1$ and $z_2$ be two distinct complex numbers and let
$\,z=(1-t)z_1 +tz_2\,$ for some real number $t$ with $0<t<1$. Then we have to prove
$$\begin{vmatrix} z-z_1 & \overline{z}-\overline{z_1} \\
z_2-z_1 & \overline{z_2}-\overline{z_1}
\end{vmatrix}\;=\;0$$
I thought about it, but don't ge...
