Proper definition use in Stoke's theorem Let the curve C be a piecewise smooth and simple closed curve enclosing a region, D.

Some sources asserts Stoke's theorem to be:

$$\oint_{C} F.dr = \iint_{R}\nabla \times FdS$$

Whereas, some claims it to be 

$$\oint_{C} F.dr = \iint_{R}\nabla \times F.n.dS$$

Could someone clear the air as to which of the above definition used is correct?

Thanks in advance.
From the first of the statements, the surface integral is written as $$\iint_R \nabla\times \vec F \cdot d\vec S$$ where I have added overarrows to clarify vector quantities. In particular, $d\vec S$ means integrating over the surface in the direction of the unit normal. 

In the second statement the surface integral is written as 

$$\iint_R \nabla\times\vec F \cdot \hat n \ dS$$

where $\hat n$ is the unit normal and the integral now is over the surface treated as a 'scalar'. Loosely speaking, $d\vec S = \hat n \, dS$.

In other words, the two statements properly understood are equivalent.
From the first of the statements, the surface integral is written as $$\iint_R \nabla\times \vec F \cdot d\vec S$$ where I have added overarrows to clarify vector quantities. In particular, $d\vec S$ means integrating over the surface in the direction of the unit normal. 

In the second statement the surface integral is written as 

$$\iint_R \nabla\times\vec F \cdot \hat n \ dS$$

where $\hat n$ is the unit normal and the integral now is over the surface treated as a 'scalar'. Loosely speaking, $d\vec S = \hat n \, dS$.

In other words, the two statements properly understood are equivalent.