I have difficulty in solving the problem 17 of chapter 11 in GTM218 "Introdution to Smooth Mannifolds, John M.Lee"
Let $T^n\subset\mathbb{C}^n$ be an n torus. Let $\gamma_j:[0,1]\rightarrow T^n$ by $\gamma_j(t)=(1,\dots,e^{2\pi it},\dots,1)$ (with $e^{2\pi it}$ in the j-th place)
Show that a closed covecter field $\omega$ on $T^n$ is exact if and only if $\int_{\gamma_j}\omega=0$ for all j from 1 to n.
According to the hint given, I am tring to prove that the pull back form $(\epsilon^n)^*\omega$ in $\mathbb{R}^n$ is exact by proving it conservative, where $\epsilon^n: \mathbb{R}^n\rightarrow T^n$ is a smooth coving map $\epsilon^n(x^1,\dots,x^n)=(e^{2\pi ix^1},\dots,e^{2\pi ix^n})$.I deduce from the hypothesis that if $\omega=\omega^idx_i$$$\int^1_0\omega^j(1\dots,e^{2\pi it},\dots,1)dt=0$$ But what i need is $$\int^b_a\sum_jw^j(e^{2\pi r^1(t)},\dots,e^{2\pi r^n(t)})\dot{r}^j(t)dt=0$$ It doesn't seem trivial to me that I can prove the later one by the former one. And I have no idea how to use the condition that $\omega$ is closed.
Here is a another solution of a similar question (more general) in stackexchange, where a proposition being apply and I don't know the reason why. That is, a closed k-form $\omega$ is in the same cohomology class with a k-form with constant coefficient under some particular basis.
Any comments are appreciated!
Acturally, the $(\epsilon^n)^*\omega$ is exact since it's close as a pull-back of a close form and applying the Poincare lemma of covector fields.