Stokes' Theorem Different Representations? $\iint_S (\nabla \times \mathbf{F}) \cdot \hat{n} \ dS$, $\iint_S ( \nabla \times \mathbf{F} ) \cdot \ dS$?

Question

I've seen Stokes' theorem written as $\oint \mathbf{F} \cdot \ d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \hat{n} \ dS$ and $\oint \mathbf{F} \cdot \ d\mathbf{r} = \iint_S ( \nabla \times \mathbf{F} ) \cdot \ dS$. I was under the impression that the first one, $\oint \mathbf{F} \cdot \ d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot \hat{n} \ dS$, is the correct one, but I've seen the other one multiple times now, so I'm wondering if I misunderstood something?

Are these two equivalent? If so, then where is the normal vector $\hat{n}$ in the second one? If not, then why, and what am I misunderstanding?

I would greatly appreciate it if people could please take the time to clarify this.

Answer 1

Physicists like to use the notation $\operatorname d \vec S$, i.e. they assign a direction to the area element, so it becomes a vector and the scalar product is "well-defined". This direction is simply the unit normal. Hence $\operatorname d \vec S= \hat n \operatorname d S$. I guesse some authors simply ommit the arrows as usual in the mathematics literature.

In summary, this are equivalent notations.