Surface Integrals: Flux, Scalar Integrals & Worked Examples

How much sunlight passes through a curved solar panel? How much fluid flows out through the surface of a balloon? These questions ask about a quantity crossing or spread over a curved two-dimensional surface — and answering them means adding up contributions over the whole surface, patch by tiny patch.

That is what a surface integral does. The scalar version, &iint;S f dS, adds up a function f over the surface (for example, the mass of a thin shell with variable density). The vector version, &iint;S F·n dS — called the flux — measures how much of the vector field F flows through the surface in the direction of the unit normal n. You compute it by writing the surface as z = g(x, y) (or parametrizing it) and converting it into an ordinary double integral over a flat region.

Key Concepts

• Flux = &iint;S F·n dS: net flow of F through S; positive means flow with the chosen normal.
• Graph surface: for z = g(x, y), the (upward) normal element is n dS = (−gx, −gy, 1) dA.
• Scalar surface integral: &iint;S f dS = &iint;D f(x, y, g(x,y)) √(1 + gx² + gy²) dA.
• Orientation matters: flipping n flips the sign of the flux.
• Closed surfaces: &iint;S F·n dS with outward n gives total outflow — the quantity in the Divergence Theorem.

Formulas

• Flux through z = g(x, y): &iint;D F(x, y, g(x,y))·(−gx, −gy, 1) dA
• dS = √(1 + gx² + gy²) dA for a graph.
• For parametrized surfaces, the normal element is ru × rv du dv.

Worked Example

Find the upward flux of F(x, y, z) = (0, 0, z) through the paraboloid z = x² + y² with z ≤ 1.

1. Surface as a graph. g(x, y) = x² + y²; gx = 2x, gy = 2y.
2. Upward normal element: n dS = (−2x, −2y, 1) dA.
3. Field on the surface. F = (0, 0, x² + y²). Dot product: F·(−2x, −2y, 1) = x² + y².
4. Project onto the unit disk D (x² + y² ≤ 1): flux = &iint;D (x² + y²) dA.
5. Polar coordinates: ∫02π ∫01 r² · r dr dθ = 2π [r⁴/4]01 = 2π/4 = π/2.
6. Answer: π/2 ≈ 1.571 — net upward flow through the bowl.

Common Misconceptions

• “Flux is just &iint; F dS.” Without the dot product with n you are integrating a vector, not the flow through the surface — F·n is essential.
• “Orientation does not matter.” Reversing the normal changes the sign; for closed surfaces the convention is outward.
• “dS equals dA.” Only for flat horizontal surfaces — for a tilted or curved patch, dS = √(1 + gx² + gy²) dA stretches the area.

Real-World Applications

• Fluid dynamics: flow rate through a pipe cross-section or a membrane.
• Electromagnetism: electric flux (Gauss’s law) and magnetic flux through surfaces.
• Heat transfer: heat flowing out through the surface of an object.
• Computer graphics: light energy arriving at a curved surface in the rendering equation.

A surface integral moves integration off flat regions onto curved ones: write the surface as a graph or parametrization, build the normal element n dS, and finish with an ordinary double integral. For flux, the dot product F·n picks out exactly the component crossing the surface — everything else slides along it and contributes nothing.