Green, Gauss & Stokes Theorems

The integral theorems connect boundary integrals with interior integrals. Learn Green's theorem in the plane, the Gauss divergence theorem, Stokes' theorem, their conditions and orientation rules, physical meaning (sources, circulation), and applications in fluid flow, heat transfer and electromagnetism.

These theorems are multidimensional versions of the fundamental theorem of calculus: what happens inside a region is determined by what crosses its boundary. They turn Maxwell's equations from integral to differential form and underpin conservation laws in fluid and heat transfer.

An analogy

The divergence theorem is like counting people in a building: the net number leaving through all doors equals the number created inside (sources) minus those absorbed (sinks).

Conditions

Fields must be continuously differentiable on the region; regions must be bounded with piecewise-smooth boundaries. Positive orientation: counter-clockwise (region on the left); outward normals for closed surfaces; right-hand rule relates C and n in Stokes.

Green's theorem

Converts a line integral around a closed curve into a double integral — used for areas (planimeters), circulation, and in 2D flow.

Divergence theorem

Net outward flux equals total source strength inside. Gauss's law ∯E·dA = Q/ε₀, conservation of mass ∂ρ/∂t + ∇·(ρv) = 0 and heat balances all come from it.

Stokes' theorem

Circulation around a loop equals the flux of curl through any surface bounded by it — Faraday's law (EMF from changing flux) and Ampère's law. Choose the easiest surface with the same boundary.

Formulas

• Green: ∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dA
• Gauss (divergence): ∯_S F · n dS = ∭_V ∇·F dV
• Stokes: ∮_C F · dr = ∬_S (∇×F) · n dS
• Area by Green: A = ½ ∮ (x dy − y dx)

Worked example: Green's theorem

Given: ∮ (x²y dx + x dy) around the unit square [0, 1]²

Find: Value.

• ∂Q/∂x − ∂P/∂y = 1 − x²
• ∬ (1 − x²) dA = 1 − 1/3

Answer: 2/3

Worked example: Stokes' theorem

Given: F = (−y, x, 0), C: circle radius 2 in the xy-plane

Find: ∮ F·dr.

• ∇×F = (0, 0, 2)
• Flux through disc = 2 × π(2)²

Answer: 8π ≈ 25.1

Common mistakes

• Wrong: Clockwise orientation in Green's theorem. — Right: Counter-clockwise (region on the left) is positive.
• Wrong: Applying Gauss to open surfaces. — Right: Close the surface or use Stokes.

Applications

• Electromagnetics: Gauss's, Faraday's and Ampère's laws.
• Fluid dynamics: Continuity and momentum balances.
• Heat transfer: Energy balances on control volumes.

Key takeaways

• Green: ∮(P dx + Q dy) = ∬(Q_x − P_y).
• Gauss: flux out = ∭ div F.
• Stokes: circulation = ∬ curl F · n.
• Mind orientation and smoothness.

Going further

• Generalised Stokes: ∫_∂Ω ω = ∫_Ω dω.
• Integral vs differential laws: Deriving PDEs from balances.
• Finite-volume method: Numerical application of Gauss's theorem.