Surface Integrals

The central idea

For a scalar field $g$ on a parametrized surface $S$, $\iint_Sg\,dS=\iint_Ug(\mathbf r(u,v))\|\mathbf r_u\times\mathbf r_v\|dudv$. For an oriented vector field, flux is $\iint_S\mathbf F\cdot\mathbf n\,dS=\iint_U\mathbf F(\mathbf r)\cdot(\mathbf r_u\times\mathbf r_v)dudv$, with cross-product order chosen for orientation.

Definitions, hypotheses, and notation

For a graph $z=f(x,y)$ with upward orientation, a nonunit oriented area vector is $\langle-f_x,-f_y,1\rangle dA$. This often avoids normalizing and then multiplying by $dS$; the normalization factors cancel. Downward orientation negates the vector. Scalar integrals still use its norm.

Flux through an open surface depends on the chosen side and does not generally measure a net source until the surface is closed. Adding a cap can enable the Divergence Theorem, after which cap flux must be subtracted with compatible orientations.

Conceptual meaning

A scalar surface integral weights actual area, while flux counts the normal component of a vector field through oriented area. Tangential field components contribute no flux. Reversing orientation negates flux but leaves scalar area integrals unchanged.

A dependable method and decision rule

1. Distinguish scalar accumulation from oriented vector flux.
2. Parametrize the surface once and compute its oriented area vector.
3. Substitute surface coordinates into the integrand or field.
4. Use a norm for scalar $dS$ and a signed cross product for flux.
5. Check orientation, boundary pieces, and symmetry before integrating.

Fully worked example

Common mistakes and why they fail