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.

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Please, help with part b)!