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Math Help - 2-manifold.Parametrization/Integration (please, help with part b)

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    2-manifold.Parametrization/Integration (please, help with part b)

    Let M be the 2- manifold in \mathbb{R}^3 consisting of all points x such that
    4x^2+y^2+4z^2 =4
    and y\ge 0


    Then $\partial M$ is the circle consisting of all points such that
    x^2+z^2=1 and y=0


    The map \alpha: D^2 \rightarrow \mathbb{R}^3, where D^2 = \{(u,v) \in \mathbb{R}^2| u^2+v^2 <1<br />
\} given by
    \alpha(u,v) = (u, 2(1-u^2-v^2)^{\frac{1}{2}},v)


    is local parametrization on M that covers M \setminus \partial M. Orient M so that \alpha belongs to the orientation, and give \partial M the induced orientation.



    a)What normal vector corresponds to the orientation of \partial M?


    b)Let \omega be the 1- form
    \omega = ydx+3xdz
    Evaluate
    \int_{\partial M} \omega directly.


    c)Evaluate \int_{M} d\omega directly , by expressing it as an integral over the unit disk D^2 in the (u,v) plane.

    -----------------------------
    Please, help with part b)!
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  2. #2
    Super Member Rebesques's Avatar
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    Re: 2-manifold.Parametrization/Integration (please, help with part b)

    Quote Originally Posted by vercammen View Post





    b)Let \omega be the 1- form
    \omega = ydx+3xdz
    Evaluate
    \int_{\partial M} \omega directly.


    c)Evaluate \int_{M} d\omega directly , by expressing it as an integral over the unit disk D^2 in the (u,v) plane.

    -----------------------------
    Please, help with part b)!

    We have \partial M=\{(x(t),y(t),z(t))=(\cos(t),0,\sin(t)):0\leq t\leq 2\pi \} and so
    \int_{\partial M} \omega=\int_0^{2\pi}(y(t)x'(t)+3x(t)y'(t))dt=0.

    Now for part c), I'd say use the Gauss theorem:
    \int_{\partial M} \omega=\int_M d\omega.
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