The $3-$ball ${B_R}^3 = \{(u,v,w) \in \mathbb{R}^3 | u^2+v^2+w^2 \le R^2\}$ is a $3-$manifold in $\mathbb{R}^3$; orient it naturally and give ${S_R}^2 = \partial {B_R}^3 = \{ (u,v,w)\in \mathbb{R}^3 | u^2+v^2+w^2 = R^2\}$ the induced orientation. Assume that $\omega$ is a $2-form$ defined in $\mathbb{R}^2 \setminus \{0\}$ such that $\int_{{S_R}^2} \omega = a+\dfrac{b}{R} $for each $R>0$,
a) Given $0
b) If $d\omega =0$, what can you say about $a$ and $b$?
c) If $\omega = d\eta$ for some $\eta$ in $\mathbb{R}^3 \setminus \{0\}$, what can you say about $a$ and $b$?
*Munkres. Chapter 7. Paragraph 37. Problem 5.
Just finished part a)
Need help with b) and c)
Thanks is advance!