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Math Help - 3-maniflod in R^3

  1. #1
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    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<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.
    Last edited by vercammen; December 20th 2012 at 07:33 AM.
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  2. #2
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    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
    Thanks from vercammen
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  3. #3
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    Re: 3-maniflod in R^3

    Thank you so much!
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