1. ## 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 $D^2 = \{(u,v) \in \mathbb{R}^2| u^2+v^2 <1
\}$
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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2. ## Re: 2-manifold.Parametrization/Integration (please, help with part b)

Originally Posted by vercammen

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