surface interal with parameters can be expressed like:
$ \iint_{S} \vec F \cdot d \vec S = \iint_D \vec F \cdot \vec n dS $
But I think that $\vec n $ can vary if using cross product because $\vec n $ can be calculated as $ \vec n = \frac{ r_u \times r_v}{\vert r_u \times r_v \vert }$, also $ \vec n = \frac{r_v \times r_v}{\vert r_v \times r_u \vert } $ , but these directions are relatively different.
How can I choose $\vec n$ from there two?