I Which Unit Normal Vector of a Surface is Correct?

[terryds]

What is actually the unit normal vector of a surface?
Is it this?

Or this one?

I see that those are opposite in direction. But, I want the correct one, which means that it should point outward.
So, which one is correct?

 

[FactChecker]

Both are unit normals. The definition of "inward" and "outward" is dependent on an entire closed region whereas both gradient and cross product are local properties. The meaning of "outward" and can not be defined locally without reference to the larger context.

 

[terryds]

Both are unit normals. The definition of "inward" and "outward" is dependent on an entire closed region whereas both gradient and cross product are local properties. The meaning of "outward" and can not be defined locally without reference to the larger context.

Is there an easy way to check it inward/outward?

 

[FactChecker]

Usually these are used in a context of integration over a surface where both the surface and the integration are defined in such a way that keeps track of outward.

 

[Infrared]

Is there an easy way to check it inward/outward?

The expressions you gave only determine $\hat{n}$ up to sign, but sign is what determines whether your normal vector is inward- or outward-pointing. If $u,v$ are two independent tangent vectors at a point on your surface, then swapping them gives a minus sign in your first formula for $\hat{n}$. Similarly, if your surface is the zero set of a function $g$, then it is also the zero set of $-g$. But swapping $g$ with $-g$ gives a minus sign in your second formula.

 

[WWGD]

A continuous, strictly inward- or outward- normal ( when given the right context, as FactChecker stated) exists only when the surface is orientable; some actually use its existence as the definition for orientability. Notice, e.g., a normal vector field on the Mobius Strip, and how it must make a discontinuous turn at some point.