# 3-maniflod in R^3

• Dec 20th 2012, 07:27 AM
vercammen
3-maniflod in R^3(Munkres. Chapter 7. Paragraph 37. Problem 5)
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, let $M$ be the $3-$ manifold in $\mathbb{R}^3$ consisting of all $x$ with $c\le ||x|| \le d$, oriented naturally. Find $\int_{M} d\omega$.

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.
• Dec 20th 2012, 12:08 PM
xxp9
Re: 3-maniflod in R^3
a) $\int_M d\omega=\int_{\partial M}\omega=(\int_{S_d^2}-\int_{S_c^2}}) \omega = (a+\frac{b}{d})-(a+\frac{b}{c})=b(\frac{1}{d}-\frac{1}{c})$
b) If $d\omega=0$, $0=\int_M d\omega=b(\frac{1}{d}-\frac{1}{c})$, so b=0
c) If $\omega=d\eta$, $\int_{S_R^2} \omega=\int_{S_R^2} d\eta = \int_{\partial S_R^2} \eta = \int_{\emptyset} \eta = 0$, so a=b=0
• Dec 20th 2012, 03:16 PM
vercammen
Re: 3-maniflod in R^3
Thank you so much!