3-maniflod in R^3

**vercammen** · December 20th 2012, 06:27 AM

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 $0<c<d$ , 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.

**xxp9** · December 20th 2012, 11:08 AM

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

**vercammen** · December 20th 2012, 02:16 PM

Thank you so much!

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

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