Surface Integrals — First and Second Kind
Big Picture: You already know how to integrate along curves (line integrals). Now we lift the dimension by one: we integrate over surfaces sitting in three-dimensional space. Two flavours arise — one that weighs a surface by a scalar density (the first kind), and one that measures how much of a vector field flows through a surface (the second kind, called flux).
1The First Kind — Scalar Surface Integrals
1.1 The Pattern You Already Own
Think of a thin curved shell — a lampshade, a dome, a piece of eggshell — with a known areal mass-density $\sigma(x,y,z)$ (mass per unit area). To find the shell's total mass you must add up $\sigma\,dS$ over every tiny patch of the surface. This is exactly the pattern you met four times already:
| Object | Density type | Element | Integral gives |
|---|---|---|---|
| Wire (1-D) | linear $\lambda$ | $ds$ | $\int_C \lambda\,ds$ |
| Flat plate (2-D) | areal $\sigma$ | $dA$ | $\iint_R \sigma\,dA$ |
| Solid body (3-D) | volume $\rho$ | $dV$ | $\iiint_E \rho\,dV$ |
| Curved shell (2-D in 3-D) | areal $\sigma$ | $dS$ | $\displaystyle\iint_S \sigma\,dS$ |
The only new ingredient is the surface area element $dS$ — the rest of the construction is the same pattern recurring.
1.2 The Surface Described Parametrically
Recall from your study of mappings: a parametric surface is a function
$$\mathbf{r}(u,v) = \bigl\langle x(u,v),\; y(u,v),\; z(u,v)\bigr\rangle, \qquad (u,v)\in D\subseteq\mathbb{R}^2$$