MAST20029 · Engineering Mathematics
Line, Work and Surface Integrals; Conservative Fields
This chapter integrates along curves and over surfaces. You parametrise a path r(t) (orientation by increasing t), then compute scalar line integrals ∫_C f ds and work integrals ∫_C F·dr; for surfaces you compute scalar surface integrals (area, mass) and flux ∬_S F·n̂ dS. The high-value idea is the conservative field: if curl F = 0 then F = ∇φ, line integrals depend only on endpoints, and the work around any closed loop is zero — letting you skip a long calculation entirely.
What this chapter covers
- 011. Parametrising a path: r(t) = (x(t), y(t), z(t)), orientation by increasing t
- 022. Scalar line integral ∫_C f ds = ∫ f(r(t)) |r'(t)| dt
- 033. Work integral ∫_C F·dr = ∫ (F₁ x' + F₂ y' + F₃ z') dt
- 044. Conservative test: curl F = 0 on a simply connected domain ⇔ F = ∇φ
- 055. Finding the potential φ by integrating F₁, F₂, F₃ and matching
- 066. Path independence: ∫_C F·dr = φ(end) − φ(start); zero around a closed loop
- 077. Scalar surface integral ∬_S g dS = ∬_R g √(f_x² + f_y² + 1) dA (g=1 gives area)
- 088. Flux integral with upward normal: ∬_S F·n̂ dS = ∬_R (−F₁ f_x − F₂ f_y + F₃) dA
Conservative field: potential and zero work around a loop
- 2 marks(a) Test curl F = 0. Check the mixed partials: ∂(2xz)/∂y = 0 = ∂(x²)/∂z; ∂(2xy+z²)/∂z = 2z = ∂(2xz)/∂x; ∂(x²)/∂x = 2x = ∂(2xy+z²)/∂y. All three pairs match, so curl F = 0 and F is conservative on this simply connected domain.
- 1 mark(b) Integrate φ_x = 2xy + z² in x: φ = x²y + xz² + g(y,z).
- 1 markMatch the other components: φ_y = x² requires g_y = 0, and φ_z = 2xz requires g_z = 0, so g is constant: φ = x²y + xz² (+C).
- 2 marks(c) The path is a CLOSED loop and F is conservative, so the work is φ(end) − φ(start) = 0 — start and end coincide. No need to integrate the four edges.
Key terms
- Parametrised path
- A curve C written as r(t) = (x(t), y(t), z(t)) for t in [a,b], oriented by increasing t. A clockwise vs anticlockwise choice flips a sign — a recurring exam subtlety.
- Work integral
- ∫_C F·dr = ∫ₐᵇ (F₁ x' + F₂ y' + F₃ z') dt, the work done by a force field F along the path C. For a conservative field it equals φ(end) − φ(start).
- Conservative field
- A field with curl F = 0 on a simply connected domain, equivalently F = ∇φ. Its line integrals are path-independent and its work around any closed loop is zero.
- Potential function φ
- A scalar φ with ∇φ = F. Found by integrating one component of F and matching the rest; it makes ∫_C F·dr = φ(end) − φ(start).
- Scalar surface integral
- ∬_S g dS = ∬_R g √(f_x² + f_y² + 1) dA for a surface z = f(x,y) over R. With g = 1 it gives the surface area; with a density it gives mass.
- Flux integral
- ∬_S F·n̂ dS, the net flow of F through a surface. For an upward normal over z = f(x,y) it equals ∬_R (−F₁ f_x − F₂ f_y + F₃) dA.
Line, Work and Surface Integrals; Conservative Fields FAQ
How do I test whether a field is conservative?
Compute the curl: on a simply connected domain, F is conservative if and only if curl F = 0. In two dimensions this reduces to ∂F₂/∂x = ∂F₁/∂y. If the curl is zero you can find a potential φ with F = ∇φ; if it is non-zero the field is not conservative and you must integrate directly.
How do I find the potential φ?
Integrate one component — say φ_x = F₁ with respect to x — which gives φ up to an unknown function of the other variables. Then differentiate your φ and match it to F₂ and F₃ to pin down that leftover function. The result is φ (plus an arbitrary constant).
Why is the work around a closed loop zero for a conservative field?
Because the work equals φ(end) − φ(start), and on a closed loop the start and end points coincide, so the difference is zero. This is the payoff of conservativity: when a question asks for work around a closed curve and the field is conservative, you quote zero rather than integrating each edge.
What is the difference between a scalar surface integral and a flux integral?
A scalar surface integral ∬_S g dS sums a scalar quantity over the surface — with g = 1 it gives surface area, with a density it gives mass. A flux integral ∬_S F·n̂ dS sums the component of a vector field through the surface — the net flow across it. They use different formulas, so read the question carefully.
Exam move
Always start a curve or surface question by writing the parametrisation or the surface formula explicitly. For work integrals, before integrating, ask whether the field is conservative — a quick curl test can replace a long calculation, and around a closed loop a conservative field gives zero work for free. Practise finding potentials by the integrate-then-match routine until it is mechanical. For surfaces, keep the two formulas distinct: the √(f_x²+f_y²+1) factor for scalar/area integrals versus the (−F₁ f_x − F₂ f_y + F₃) form for upward flux. Watch path orientation — a clockwise loop flips a sign — and state every result used so the marker can follow the reasoning.
