How to know the direction of the unit normal vector to an open surface? For an open surface shape (assume surface $x=4$) there is 2 possibilities for the unit normal vector direction either $(+î)$ or $(-î)$. so how can we know what direction it is supposed to be (ie $.. - î$ or $+î$)?
 A: The unit normal vector is part of the definition of a surface. To completely define a surface you must give (choice) its unit normal vector. Consequently you must be careful using it for calculations. For example the flux of a vector function for one choice would be the opposite of that of the other choice etc.

In Figure-01 above we see a surface in $\:\mathbb R^3\:$ represented by the parametric equation $\:\mathbf x\left(u,v\right)$. Note that $\:\mathbf x_{u}\boldsymbol{=}\partial \mathbf x/\partial u\:$ and $\:\mathbf x_{v}\boldsymbol{=}\partial \mathbf x/\partial v\:$ are  tangent vectors to the parametric curves,  $\mathbf{x}_{u}\boldsymbol{\times}\mathbf{x}_{v}\boldsymbol{\ne 0}$ and the unit normal vector shown is
\begin{equation}
\mathbf N \boldsymbol{=} \dfrac{ \mathbf{x}_{u}\boldsymbol{\times}\mathbf{x}_{v}}{\Vert \mathbf{x}_{u}\boldsymbol{\times}\mathbf{x}_{v}\Vert}
\tag{01}\label{01}
\end{equation}
But you could equally well choose as unit normal vector $\mathbf N' \boldsymbol{=-}\mathbf N$.
