But when you view things this way — we just saw that this entire thing simplifies to dS — then our whole surface integral can be rewritten, instead of writing it like this, we can write it as ∫ ∫ F · (∂r u × ∂r v), the surface integral — those integral signs were too fancy — the surface integral of F, dot, instead of saying a normal vector times a scalar quantity, that little chunk of area on the surface, we can now just call that the vector differential dS. 