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.

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Please, help with part b)!