Unit Normal Field on a 2D manifold installed in R ^ 3

Let us think that we are offered a shut, orientable 2D manifold installed in $R^3$, and also allows call it $M$.

I assume it is clear that in a coordinate area $(U, \phi)$ it is feasible to at each factor specify a device regular vector $N_p$ by considering the tangent room $T_pM$ as a subspace of $T_pR^3$ and also taking into consideration the 1D vector room $T_pM^\perp$. Nonetheless, there are constantly 2 selections of $N_p$.

With ease, an alignment needs to need to specify $N_p$ distinctly, yet I am not exactly sure just how to make use of that $M$ is oriented to do this. If we have an alignment 2 - kind $\Omega$ on $M$, just how can we utilize this to identify which of both feasible $N_p$ is is proper?

In addition, given that we have actually $N_p$ specified at each factor in $U$, we have a vector area on $U$. I was not exactly sure just how to show that this is a smooth vector area, yet I think that as soon as you've done this, you can in some way make use of a dividing of unity to expand $N$ to the whole manifold. Yet, I assume that there is some nuance also to this.

I suggest, for a non - orientable manifold such as a Möbius strip, it appears to me that you can specify in your area a smooth device - regular vector area, yet when you attempt to expand it to the entire manifold it is no more a continual vector area, so in some way the alignment has to contribute in the expansion, and also I am not exactly sure just how.

2019-12-02 02:50:13
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Answers: 1

There is a typical device quantity kind $\omega = dx\wedge dy\wedge dz$ on $\wedge R^3$. Getting with repect to a device vector $v$ offers a 2 - kind $\omega'$. Drawing back to your surface area if $v$ is regular to $M$ at $p$ after that $\omega'$ draws back to a multiple of $\Omega$ at $p$. If this is a favorable numerous after that $v$ and also $\Omega$ are sympathetically oriented.

2019-12-03 04:25:02