Line & Surface Integrals
Line integrals add up quantities along curves; surface integrals add them over surfaces. Learn scalar and vector line integrals, work done by a force field, conservative fields, path independence and potential functions, scalar surface integrals, flux through surfaces, orientation, and closed surfaces.
The work done by a varying force along a path, the circulation of air around a wing, the flow of water through a net, and the electric flux through a surface are all line or surface integrals.
An analogy
A line integral of force is like measuring the effort of walking a winding trail against the wind; a flux integral is like counting how much rain falls through a tilted net.
Parametrise the curve
Write r(t) for the path, compute r′(t), substitute into F, and integrate over t. Direction matters: reversing the path changes the sign.
Conservative fields
Gravity and electrostatic fields are conservative: work depends only on endpoints, and around closed loops it is zero. Test with curl F = 0 and find φ by integrating components.
Flux
Flux measures how much of a field passes through a surface, using the unit normal n (orientation matters — outward for closed surfaces). Electric flux, mass flow rate, heat flow through a wall.
Non-conservative fields
Magnetic fields around currents and friction forces are path-dependent; ∮ F · dr ≠ 0 measures circulation — the idea behind Ampère's law and lift on aerofoils.
Formulas
- Vector line integral:
∫_C F · dr = ∫ₐᵇ F(r(t)) · r′(t) dt - Conservative fields:
F = ∇φ ⇔ ∇×F = 0 (simply connected) ⇒ ∫_C F·dr = φ(B) − φ(A) - Flux:
Φ = ∬_S F · n dS - Scalar surface integral:
∬_S f dS (area, mass of shells)
Worked example: Circulation
Given: F = (−y, x), C: unit circle counter-clockwise
Find: ∮ F · dr.
- r = (cos t, sin t), r′ = (−sin t, cos t)
- F·r′ = sin²t + cos²t = 1
- ∫₀^{2π} 1 dt
Answer: 2π (non-conservative)
Worked example: Conservative field
Given: F = (y, x) from (0, 0) to (1, 2)
Find: Work.
- ∇×F = 1 − 1 = 0 → φ = xy
- W = φ(1, 2) − φ(0, 0)
Answer: 2 (any path)
Common mistakes
- Wrong: Ignoring path direction. — Right: Reversing C changes the sign.
- Wrong: Wrong normal orientation. — Right: Use outward normals for closed surfaces.
Applications
- Mechanics: Work by variable forces along paths.
- Electromagnetics: EMF = ∮E·dl; flux Φ = ∬B·dA.
- Fluids: Mass flow through surfaces; circulation and lift.
Key takeaways
- Parametrise: ∫F(r(t))·r′(t) dt.
- Conservative: curl = 0, φ(B) − φ(A).
- Flux = ∬F·n dS.
- Orientation matters.
Going further
- Differential forms: Unified view of line/surface integrals.
- Stream functions: 2D incompressible flow.
- Numerical line integrals: Quadrature along curves.