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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)?

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    $\begingroup$ 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. $\endgroup$
    – Frobenius
    Jul 18 at 21:32
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    $\begingroup$ @Frobenius , Beautifully explained ! $\endgroup$ Jul 18 at 21:34
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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.

enter image description here

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 $\mathbf N' \boldsymbol{=-}\mathbf N$.

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    $\begingroup$ Thanks very much indeed , Mr.Frobenius $\endgroup$ Jul 18 at 22:28
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    $\begingroup$ @Carlos Werbock : Welcome to PSE. Good Luck. $\endgroup$
    – Frobenius
    Jul 18 at 22:30
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    $\begingroup$ Just only for the image it is necessary to vote up it :-) $\endgroup$
    – Sebastiano
    Jul 19 at 9:53

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