$\iint^b_a \lvert \frac {da}{db} × \frac {da}{dc}\rvert dbdc$
Pardon the subpar writing above in MathJax. I am using the surface integral 'formula' to calculate the surface area of a 3-D torus i.e. a doughnut. Could someone in 'simple' terms explain why this formula is used? How does one derive this formula? I am a senior in high school so my experience with multivariable calculus is fairly basic.
I hope you are well familiar with the cross product. In general, the surface integral of a function f over a surface $S$ is written $ \int_{S}fd\Sigma$, where $d\Sigma$ is sort of a "surface dx" pointing outwards from the surface , Without going into details, if one were to "parametrise" the surface with x(b,c) and just saying f=1 we would have the area, and then $\Sigma$ would be the derivatives of a with respect of b and c, and taking the cross product of these followed by magnitude will give us something perpendicular to the surface, which is what we want.
Happy New Years!