Mixing
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Lift on non star-shaped domain
$begingroup$
I came across the following exercise.
Let
$$
A = { x in mathbb{R}^2 : 1 leq lvert x rvert leq 2 }.
$$
Show that there is no continuous function $vartheta: A rightarrow mathbb{R}$ such that
$$
E(x) := frac{x}{lvert x rvert} = (cosvartheta(x), sin vartheta(x)) quad text{for all} ; x in A. quad quad (*)
$$
I know that $(*)$ would hold, if $A$ was star-shaped and $E$ a continuous unit vector field.
However, I am not able to find a contradiction.
complex-analysis analysis differential-geometry logarithms
asked Jan 27 at 19:01
fpmoofpmoo
382113
$endgroup$
add a comment |
$begingroup$
I came across the following exercise.
Let
$$
A = { x in mathbb{R}^2 : 1 leq lvert x rvert leq 2 }.
$$
Show that there is no continuous function $vartheta: A rightarrow mathbb{R}$ such that
$$
E(x) := frac{x}{lvert x rvert} = (cosvartheta(x), sin vartheta(x)) quad text{for all} ; x in A. quad quad (*)
$$
I know that $(*)$ would hold, if $A$ was star-shaped and $E$ a continuous unit vector field.
However, I am not able to find a contradiction.
complex-analysis analysis differential-geometry logarithms
asked Jan 27 at 19:01
fpmoofpmoo
382113
$endgroup$
add a comment |
$begingroup$
I came across the following exercise.
Let
$$
A = { x in mathbb{R}^2 : 1 leq lvert x rvert leq 2 }.
$$
Show that there is no continuous function $vartheta: A rightarrow mathbb{R}$ such that
$$
E(x) := frac{x}{lvert x rvert} = (cosvartheta(x), sin vartheta(x)) quad text{for all} ; x in A. quad quad (*)
$$
I know that $(*)$ would hold, if $A$ was star-shaped and $E$ a continuous unit vector field.
However, I am not able to find a contradiction.
complex-analysis analysis differential-geometry logarithms
asked Jan 27 at 19:01
fpmoofpmoo
382113
$endgroup$
I came across the following exercise.
Let
$$
A = { x in mathbb{R}^2 : 1 leq lvert x rvert leq 2 }.
$$
Show that there is no continuous function $vartheta: A rightarrow mathbb{R}$ such that
$$
E(x) := frac{x}{lvert x rvert} = (cosvartheta(x), sin vartheta(x)) quad text{for all} ; x in A. quad quad (*)
$$
I know that $(*)$ would hold, if $A$ was star-shaped and $E$ a continuous unit vector field.
However, I am not able to find a contradiction.
complex-analysis analysis differential-geometry logarithms
complex-analysis analysis differential-geometry logarithms
asked Jan 27 at 19:01
fpmoofpmoo
382113
asked Jan 27 at 19:01
fpmoofpmoo
382113
edited Jan 27 at 19:14
fpmoo
asked Jan 27 at 19:01
fpmoofpmoo
382113
asked Jan 27 at 19:01
fpmoofpmoo
382113
asked Jan 27 at 19:01
fpmoofpmoo
382113
382113
add a comment |
add a comment |
1 Answer
1
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Hint: Note $e^{it} = e^{ivartheta (e^{it})}.$ Thus $vartheta (e^{it}) = t +2pi n_t,$ where $n_tin mathbb Z.$
answered Jan 27 at 19:22
zhw.zhw.
74.7k43175
$endgroup$
add a comment |
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$begingroup$
Hint: Note $e^{it} = e^{ivartheta (e^{it})}.$ Thus $vartheta (e^{it}) = t +2pi n_t,$ where $n_tin mathbb Z.$
answered Jan 27 at 19:22
zhw.zhw.
74.7k43175
$endgroup$
add a comment |
$begingroup$
Hint: Note $e^{it} = e^{ivartheta (e^{it})}.$ Thus $vartheta (e^{it}) = t +2pi n_t,$ where $n_tin mathbb Z.$
answered Jan 27 at 19:22
zhw.zhw.
74.7k43175
$endgroup$
add a comment |
$begingroup$
Hint: Note $e^{it} = e^{ivartheta (e^{it})}.$ Thus $vartheta (e^{it}) = t +2pi n_t,$ where $n_tin mathbb Z.$
answered Jan 27 at 19:22
zhw.zhw.
74.7k43175
$endgroup$
Hint: Note $e^{it} = e^{ivartheta (e^{it})}.$ Thus $vartheta (e^{it}) = t +2pi n_t,$ where $n_tin mathbb Z.$
answered Jan 27 at 19:22
zhw.zhw.
74.7k43175
answered Jan 27 at 19:22
zhw.zhw.
74.7k43175
answered Jan 27 at 19:22
zhw.zhw.
74.7k43175
answered Jan 27 at 19:22
zhw.zhw.
74.7k43175
74.7k43175
add a comment |
add a comment |
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