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.