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Thread: 3-maniflod in R^3

  1. #1
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    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.
    Last edited by vercammen; Dec 20th 2012 at 06:33 AM.
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  2. #2
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    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
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  3. #3
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    Re: 3-maniflod in R^3

    Thank you so much!
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