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15h
comment Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$
Am I correct .. ?
15h
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.
15h
comment Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$
Integral path independence?
1d
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?
1d
asked Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$
Jul
19
accepted One-to-one continuous mapping preserve openness?
Jul
19
accepted Continuous derivative vs Continuous partial derivatives
Jul
18
comment Continuous derivative vs Continuous partial derivatives
Actually $f'$ is a also R^n to R^m fucntion, i.e. like a m*n matrix.
Jul
18
asked Continuous derivative vs Continuous partial derivatives
Jul
18
comment One-to-one continuous mapping preserve openness?
I searched about this theorem, and seems that its proof is very complicated. I just added the extra condition $detg'(x)\neq 0$ for all $x\in U$, together with the differntiability, is there a simpler way to show that $g(U)$ is open?
Jul
18
revised One-to-one continuous mapping preserve openness?
added 42 characters in body
Jul
18
revised One-to-one continuous mapping preserve openness?
deleted 10 characters in body
Jul
18
asked One-to-one continuous mapping preserve openness?
Jul
17
comment How to explain your area of study to non-math people
You do not trully understand something until you can explain it to your grandmother - Albert Einstein
Jul
13
asked $f$ integrable vs $\int_Af$ exist - in Spivak's Calculus on Manifolds
Jul
10
comment Partition of Unity in Spivak's Calculus on Manifolds
I am reading calculus on manifolds too. And as I came across this proof today, I have exactly the same doubt. Glad that I find this post.
Jun
24
accepted How to define the set $\lim_{n\to\infty}\{1/n,2/n,…,n/n\}$ rigorously?
Jun
7
accepted Is this a valid partial fraction decomposition?
Jun
5
revised Is this a valid partial fraction decomposition?
added 100 characters in body
Jun
5
comment Is this a valid partial fraction decomposition?
The original function is not defined for x=-1,2...