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

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

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


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


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


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



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


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


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

    -----------------------------
    Please, help with part b)!
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    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 $\displaystyle \omega$ be the $\displaystyle 1-$ form
    $\displaystyle \omega = ydx+3xdz$
    Evaluate
    $\displaystyle \int_{\partial M} \omega$ directly.


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

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

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

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