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

Graphical or geometric meaning

Each small surface patch carries a normal arrow whose length represents patch area. Dotting the field with that arrow keeps outward flow, rejects tangential flow, and assigns negative sign to inward flow.

Common mistakes and why they fail

Verification and reasonableness checks

• Reverse orientation and confirm only flux changes sign.
• Use symmetry or the Divergence Theorem for a closed surface.
• Check scalar versus vector output and physical units.

Scalar area and flux use different integrands

A scalar surface integral $\iint_S f\,dS$ uses a nonnegative area element and does not depend on orientation. Flux $\iint_S\mathbf F\cdot\mathbf n\,dS$ uses an oriented normal and changes sign when the orientation reverses. With a parameterization, scalar area uses $\|\mathbf r_u\times\mathbf r_v\|$, while flux uses the signed vector $\mathbf r_u\times\mathbf r_v$ in the chosen order. Convert every coordinate of $f$ or $\mathbf F$ to the parameters. For closed surfaces, “outward” fixes the sign; test the normal at an easy point. A flux magnitude larger than a simple bound based on maximum field size times area may reveal a missing normalization or duplicated patch.

Practice

1. Find $\iint_S1dS$ on a sphere of radius $a$.
2. What is flux of a tangent field through a surface?
3. What happens to flux under normal reversal?

Connections and next steps

• Surface Area
• Divergence Theorem
• Stokes' Theorem