Integration of Forms and Stokes' Theorem

Integration on Manifolds and Stokes' Theorem

Idea: “Proper” Integration on Curved Surfaces

The standard integral ∫∫_D f dA “lives” in the plane. How do we integrate on a curved surface? We need an object that transforms correctly under a change of coordinates — a differential form.

An n-form ω on an n-dimensional manifold M is a “density” for integration: in coordinates, ω = f(x) dx¹ ∧ ... ∧ dxⁿ. When changing coordinates, the Jacobian of the transformation is “built in” automatically — the integral is invariant.

Orientation

A manifold M is orientable if there exists an atlas with positive Jacobians of the transition functions: det(∂x/∂y) > 0. Orientation is the choice of a “consistent” direction in all charts.

Orientable: sphere S², torus T², ℝⁿ.

Non-orientable: Möbius strip (it has no “two sides”), Klein bottle.

On an orientable M, a “right-handed” basis is chosen at each point. This allows us to define a “positive” volume element and the integral.

Integral of an n-Form

On an n-dimensional orientable manifold M with chart (u, φ):

∫M ω = ∫{φ(u)} f(φ⁻¹(x)) |det(∂φ⁻¹/∂x)| dx¹...dxⁿ

The definition is correct: it does not depend on the choice of chart (the Jacobian compensates for coordinate differences).

Numerical example: Integral of the form ω = x dy ∧ dz + y dz ∧ dx + z dx ∧ dy over the unit sphere S² (a 2-form in 3 dimensions ω is a 2-form, integrated over the 2-dimensional surface).

In spherical coordinates: x = sin φ cos θ, y = sin φ sin θ, z = cos φ. ω becomes sin φ dφ ∧ dθ · R = R (sphere of unit radius). ∫{S²} ω = ∫₀^π ∫₀^{2π} sin φ dφ dθ = 4π. This matches the divergence theorem: ∫{S²} (r · n) dS = ∫_B div(r) dV = 3·(4π/3) = 4π ✓.

Generalized Stokes' Theorem

Theorem: For an (n−1)-form ω on an oriented manifold M with boundary ∂M:

∫M dω = ∫{∂M} ω

This is the single theorem from which all formulas of integral vector algebra follow:

Newton–Leibniz: M = [a, b], ∂M = {b} − {a}. ω = f(x). dω = f'(x) dx. ∫_a^b f'(x) dx = f(b) − f(a) ✓.

Green’s Theorem: M is the domain D in ℝ², ∂M = ∂D. ω = P dx + Q dy. dω = (∂Q/∂x − ∂P/∂y) dx ∧ dy. ∬D (∂Q/∂x − ∂P/∂y) dA = ∮{∂D} (P dx + Q dy) ✓.

Stokes’ Theorem (classical): M is the surface S, ∂M = ∂S. ω is a 1-form (corresponds to F). dω is a 2-form (curl of F). ∬S rot(F) · n dS = ∮{∂S} F · dr ✓.

Gauss–Ostrogradsky Theorem: M is the volume V, ∂M = ∂V. ω is a 2-form (corresponds to F). dω is a 3-form (divergence of F). ∭V div(F) dV = ∬{∂V} F · n dS ✓.

Numerical Example: Application of Stokes’ Theorem

Let us compute ∬_S rot(F) · dS for F = (y, z, x) over the hemisphere S: x²+y²+z²=1, z≥0, boundary ∂S: x²+y²=1, z=0.

Stokes’ theorem: ∬S rot(F) · dS = ∮{∂S} F · dr.

The boundary ∂S is the unit circle: r(t) = (cos t, sin t, 0), dr = (−sin t, cos t, 0) dt.

∮_{∂S} F · dr = ∫₀^{2π} (sin t, 0, cos t) · (−sin t, cos t, 0) dt = ∫₀^{2π} −sin²t dt = −π.

Instead of integrating over the surface (difficult) — contour integral (easy). Saved effort!

Degree of a Mapping

Degree of a mapping deg(f) for f: M → N between compact oriented manifolds of equal dimension n: deg(f) = ∫_M f*ω / ∫_N ω, does not depend on the choice of the n-form ω.

A topological invariant. For f: S¹ → S¹: deg(f) = winding number. f(z) = zⁿ: deg = n.

Application: In field theory, the topological charge (of the instanton) = degree of the field mapping onto the class space. Degree = integer, cannot change continuously → topological stability of solitons.

Stokes' Theorem in Physical Conservation Laws

Stokes' theorem combines all four fundamental theorems of vector analysis into a single formula. In electrodynamics, it yields Faraday’s law (the integral of E around the contour equals the flux of the derivative of B through the surface), Gauss’s theorem for E and B, and Ampère’s law with displacement current. This makes Maxwell’s equations coordinate-invariant and ready for generalization to curved spacetime. In computational mechanics, the boundary element method (BEM) replaces the volume integral in a differential equation with a boundary integral, reducing the dimensionality of the problem by one. This makes it possible to numerically solve problems in acoustics, electrostatics, and elasticity theory for bodies of complex shape much more efficiently than the finite element method. In hydrodynamics, the Gauss–Ostrogradsky theorem relates the fluid flow through a closed surface to the divergence of the velocity field — the basis of the continuity equation. In potential theory (Dirichlet and Neumann problems), Green's formula expresses the value of a harmonic function inside the domain through its values on the boundary, which is a direct consequence of Stokes' theorem and forms the foundation of the boundary integral equation method. Topological versions of Stokes’ theorem (in terms of cohomology) classify global obstructions to the existence of potentials.

Exercise: (a) Compute ∮∂D (x dy − y dx)/2 for the square [0,1]². What is this? (b) For the form ω = x dy ∧ dz on the sphere S²: calculate ∬{S²} ω by direct integration and via Stokes’ theorem (Gauss). (c) Why does Stokes’ theorem ∫M dω = ∫{∂M} ω subsume all four theorems of vector analysis? Write the schema (M, ∂M, ω) for each.