Surface Integral Differential

The surface area differential $d\sigma$ is defined differently for the type of equation you are using. The differential is often used hand-in-hand with surface’s normal vectors and even cancels terms

Parametric Surface

$$d\sigma =|r\_u\times r\_v|dudv$$

Explicit Surface

$$d\sigma =\sqrt{f\_x^2+f\_y^2+1}dxdy$$

Implicit Surface $\overrightarrow{n\_plane}$ is the normal vector to the area element $dA$ (for $dxdy$, the xy-plane it would be just $k$) not to be confused with the surface normal vector.

$$d\sigma =\frac{|\nabla f|}{|\nabla f\cdot \overrightarrow{n\_plane}|}dA$$