# 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)?

• 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. Jul 18 at 21:32
• @Frobenius , Beautifully explained ! Jul 18 at 21:34

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 $$$$\mathbf N \boldsymbol{=} \dfrac{ \mathbf{x}_{u}\boldsymbol{\times}\mathbf{x}_{v}}{\Vert \mathbf{x}_{u}\boldsymbol{\times}\mathbf{x}_{v}\Vert} \tag{01}\label{01}$$$$
But you could equally well choose as unit normal vector $$\mathbf N' \boldsymbol{=-}\mathbf N$$.