Line Integrals, Flux, and Conservative Fields
Core formulas and concepts for integrating scalar functions and vector fields along curves.
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Questions Covered in This Set
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Scalar line integral formula ∫_C f ds
∫_a^b f(r(t))·|r′(t)| dt, where ds = |r′(t)| dt. It is orientation-blind (walking the curve backwards gives the same value). Models the mass of a wire with linear density f.
Work (vector) line integral ∫_C F·dr
∫_a^b F(r(t))·r′(t) dt = ∫_C P dx + Q dy + R dz. It measures the tangential push of the field along C, and it FLIPS SIGN if you reverse orientation.
Example: mass of the helix r(t)=⟨cos t, sin t, t⟩, 0≤t≤2π, density f = z
r′=⟨−sin t, cos t, 1⟩ so |r′|=√2. ∫_0^{2π} t√2 dt = √2·(2π²)/... = 2√2 π².
Flux of a 2D field across a plane curve
∮_C F·n ds = ∮_C P dy − Q dx, where n = ⟨y′, −x′⟩/|r′| is the outward normal (tangent rotated −90°) for a counterclockwise curve.
Work vs. flux mnemonic
Work pairs with dr (tangential component); flux pairs with the rotated dr (normal component). Same field, same curve, two genuinely different measurements.
Why does F = ⟨y, −x⟩ give work −2π and flux 0 on the unit circle (CCW)?
F·r′ = −sin²t − cos²t = −1, so work = −2π (you walk against the clockwise swirl). P dy − Q dx = sin t cos t − sin t cos t = 0, so nothing is pumped across the circle — pure rotation.
Fundamental Theorem for Line Integrals
If F = ∇f, then ∫_C ∇f·dr = f(r(b)) − f(r(a)). Proof: d/dt f(r(t)) = ∇f·r′(t), then apply the ordinary FTC.
Three equivalent characterizations of a conservative field (connected domain)
(1) F = ∇f for some potential f; (2) ∫_C F·dr depends only on endpoints (path independence); (3) ∮_C F·dr = 0 around every closed loop.
Why are path independence and zero loop integrals equivalent?
Take two paths with the same endpoints, reverse one (which negates its integral), and glue them into a closed loop; the loop integral is the difference of the two, so it vanishes exactly when they agree.
Screening test for conservativeness in 2D
If F = ⟨P, Q⟩ = ∇f with f smooth, mixed partials commute, so ∂P/∂y = ∂Q/∂x. (In 3D: curl F = 0.) On a simply connected domain this condition is also sufficient.
