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Aug
26
awarded  Nice Question
Aug
20
accepted Will this punch a hole in the field of complex number?
Aug
18
revised Will this punch a hole in the field of complex number?
deleted 1 character in body
Aug
18
revised Will this punch a hole in the field of complex number?
deleted 56 characters in body
Aug
18
comment Will this punch a hole in the field of complex number?
I have added some more.
Aug
18
revised Will this punch a hole in the field of complex number?
deleted 56 characters in body
Aug
18
revised Will this punch a hole in the field of complex number?
deleted 56 characters in body
Aug
18
comment Will this punch a hole in the field of complex number?
I guess that doesnt count breaking complex number XD
Aug
18
asked Will this punch a hole in the field of complex number?
Aug
10
comment I am confused by the different definitions of manifolds.
Thank you for your detailed answer. Your definition given is just like what I usually see, and I can start smelling the homeomorphism in Carmo's definition. But there one place left unclear to me. I understand $x_\alpha$ are bijection. Also, since for all open $A\subset \textbf{x}_\alpha(U_\alpha)$, $\textbf{x}_\alpha ^{-1}(A)$ is open, so $\textbf{x}_\alpha$ are continuous. However, how can we show $\textbf{x} _\alpha^{-1}$ are also continuous?
Aug
8
asked I am confused by the different definitions of manifolds.
Aug
7
awarded  Popular Question
Aug
6
revised Proper definition use in Stoke's theorem
added 10 characters in body
Aug
3
revised Find number of element in $\{m\in\mathbb N:m\leq n\text{ and }m\text{ has the digit 3}\}$.
added 31 characters in body
Aug
3
asked Find number of element in $\{m\in\mathbb N:m\leq n\text{ and }m\text{ has the digit 3}\}$.
Aug
3
accepted Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$
Aug
2
comment Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$
Am I correct .. ?
Aug
2
comment Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$
I think I get it. From previous exercise, I showed that such $g_R$ have the property that $g_R(0)=g_R(1)$. Also, we can replace the $1$ in the hint by $n$. Therefore let $\mu=\lambda/2\pi n$, $\int _{c_{R,n}}(\omega-\mu d\theta)=\int_{[0,1]}(c_{R,n}^*\omega)-\mu 2\pi n=\int_{[0,1]}(\lambda dx+dg_R)-\mu 2\pi n=\lambda-\mu 2\pi n=0$. Path independence imply exactness.
Aug
2
comment Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$
Integral path independence?
Aug
1
comment Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$
Errm, but I can't see how this lead me closer to the answer, can you please give more hint?