1. 3-maniflod in R^3(Munkres. Chapter 7. Paragraph 37. Problem 5)

The $\displaystyle 3-$ball $\displaystyle {B_R}^3 = \{(u,v,w) \in \mathbb{R}^3 | u^2+v^2+w^2 \le R^2\}$ is a $\displaystyle 3-$ manifold in $\displaystyle \mathbb{R}^3$;
orient it naturally and give

$\displaystyle {S_R}^2 = \partial {B_R}^3 = \{ (u,v,w)\in \mathbb{R}^3 | u^2+v^2+w^2 = R^2\}$
the induced orientation. Assume that $\displaystyle \omega$ is a $\displaystyle 2-form$ defined in $\displaystyle \mathbb{R}^2 \setminus \{0\}$ such that
$\displaystyle \int_{{S_R}^2} \omega = a+\dfrac{b}{R}$ for each $\displaystyle R>0$,

a) Given $\displaystyle 0<c<d$, let $\displaystyle M$ be the $\displaystyle 3-$ manifold in $\displaystyle \mathbb{R}^3$ consisting of all $\displaystyle x$ with $\displaystyle c\le ||x|| \le d$, oriented naturally. Find $\displaystyle \int_{M} d\omega$.

b) If $\displaystyle d\omega =0$, what can you say about $\displaystyle a$ and $\displaystyle b$?

c) If $\displaystyle \omega = d\eta$ for some $\displaystyle \eta$ in $\displaystyle \mathbb{R}^3 \setminus \{0\}$, what can you say about $\displaystyle a$ and $\displaystyle b$?

*Munkres. Chapter 7. Paragraph 37. Problem 5.

2. Re: 3-maniflod in R^3

a) $\displaystyle \int_M d\omega=\int_{\partial M}\omega=(\int_{S_d^2}-\int_{S_c^2}}) \omega = (a+\frac{b}{d})-(a+\frac{b}{c})=b(\frac{1}{d}-\frac{1}{c})$
b) If $\displaystyle d\omega=0$, $\displaystyle 0=\int_M d\omega=b(\frac{1}{d}-\frac{1}{c})$, so b=0
c) If $\displaystyle \omega=d\eta$, $\displaystyle \int_{S_R^2} \omega=\int_{S_R^2} d\eta = \int_{\partial S_R^2} \eta = \int_{\emptyset} \eta = 0$, so a=b=0

3. Re: 3-maniflod in R^3

Thank you so much!