Stokes' Theorem

Stokes' theorem is one of the most powerful results in calculus: a single formula ∫_M dω = ∫_{∂M} ω unifies four classical theorems. It underlies electrodynamics, fluid mechanics, finite element methods, and algebraic topology. Every time a physicist swaps a volume integral for a surface integral, they're using Stokes.

• **Electromagnetism:** Faraday's and Ampère-Maxwell's laws are direct applications of Stokes for the electromagnetic 2-form F = E∧dt + B
• **FEM/FVM:** numerical methods use Stokes to convert volume integrals to boundary integrals, reducing integration dimension
• **Fluid dynamics:** Kelvin's circulation theorem - rate of change of circulation equals flux of vorticity - is a consequence of Stokes

Предварительные знания

• Gauss-Bonnet Theorem

The Generalized Stokes Theorem

**Generalized Stokes theorem:** for a (k−1)-form ω on a compact oriented manifold M with boundary ∂M: ∫_M dω = ∫_{∂M} ω. A single formula unifies all the classical flow and circulation theorems.

The interpretation: the integral of the derivative over the interior equals the integral of the original form over the boundary. This directly generalizes the Fundamental Theorem of Calculus: ∫_a^b f'(x) dx = f(b) − f(a), where {a, b} = ∂[a,b].

**Boundary operator ∂:** ∂² = 0 (the boundary of a boundary is empty). Exterior derivative d² = 0. Stokes' theorem shows d and ∂ are dual operators: ∫_M dω = ∫_{∂M} ω. This is the algebraic foundation of cohomology theory.

Green's Theorem, Divergence Theorem, and Classical Stokes

**Green's theorem** (∂M = closed curve in R²): ∮_C (P dx + Q dy) = ∫∫_D (∂Q/∂x − ∂P/∂y) dA. Application: area of D via a line integral S = (1/2)∮(x dy − y dx).

**Divergence theorem:** ∫∫∫_V div F dV = ∯_{∂V} F·n dA. Flux through a closed surface = volume integral of divergence. **Classical Stokes:** ∫∫_S (∇×F)·n dA = ∮_{∂S} F·dr. Circulation = flux of curl.

Applications to Physics and Numerical Methods

**Electromagnetism:** Faraday's law ∮ E·dr = −d/dt ∫∫ B·dA and Ampère-Maxwell ∮ B·dr = μ₀∫∫ J·dA + μ₀ε₀ d/dt ∫∫ E·dA are both statements of the generalized Stokes theorem for the electromagnetic 2-form F.

**Conservative fields:** F is conservative ↔ ∮_C F·dr = 0 for every closed C ↔ curl F = 0 ↔ F = −∇U (on simply connected domains). By Stokes: ∮ F·dr = ∫∫ curl F · dS = 0 if and only if curl F = 0.

In FEM and FVM, Stokes' theorem converts volume integrals (div F) to surface integrals (flux F·n), reducing the dimension of integration. Green's identity ∫∇u·∇v = ∫fv comes from Stokes and is the foundation of the weak formulation of PDEs.

Key Ideas

• **Generalized Stokes:** ∫_M dω = ∫_{∂M} ω. Integral of the derivative = boundary integral. ∂² = 0 dual to d² = 0
• **Special cases:** Newton-Leibniz (1D), Green (2D), Gauss/divergence (3D volume), classical Stokes (3D surface)
• **Conservative fields:** ∮ F·dr = 0 ↔ curl F = 0 ↔ F = −∇U (on simply connected domains)
• **Numerical use:** FEM weak formulations come from Green's identity, which is a consequence of Stokes

Related Topics

Stokes' theorem is the central result of analysis on manifolds:

• Differential Forms — ω is a (k−1)-form, dω is a k-form. Stokes works with the exterior derivative
• Gauss-Bonnet Theorem — Gauss-Bonnet follows from the generalized Stokes and Chern-Weil theory
• Functional Analysis in PDEs — Weak PDE formulations via Green's identity rely on Stokes theorem

Вопросы для размышления

• Green's theorem gives area as S = (1/2)∮(x dy − y dx). Implement this for a polygon (list of vertices) and verify it gives the shoelace formula.
• Faraday's law ∮ E·dr = −d/dt ∫∫ B·dA is a Stokes statement. What is the 2-form ω and what is dω in the spacetime formulation? How does the unified form dF = 0 encode this?
• FEM weak formulation of −Δu = f: multiply by test function v and integrate by parts to get ∫∇u·∇v = ∫fv. Which version of Stokes (Green's identity) makes this step valid?

Связанные уроки

• calc-11-definite