I understand that when calculating the surface integral, we have to parametrize the surface by a 2-variables vector equation. Then, the cross product of the two velocity partial vectors would then become the area element. This cross product, at the same time, would be the candidate of normal vector.
However, I also know that the gradient vector would be perpendicular to its level set or level surface.
So I wonder could I use the magnitude of the gradient vector as the area element?