I'm unsure how to decide whether the normal should be positive or negative in $\hat{n}dS=\pm h_2 h_3 e_1 du_2 du_3$, where $h_i$ are the scale factors, $e_i $ are the base vectors, and $u_i$ are the curvilinear coordinates.

So for example, in spherical coordinate with $r, \theta, \phi$, and $S: 0<r\leq2, \theta = \frac{\pi}{4}, 0<\phi\leq \pi $. Orientation is given as $ \hat{n} \cdot e_z > 0 $. Here how do I know that $\hat{n}dS = \pm e_\theta h_r h_\phi dr d \phi = -e_\theta \sin \theta dr d\phi $, it should be a negative?


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