Advanced undergraduateGeometry & TopologyLesson available

Differential forms and Stokes' theorem

The geometer's calculus: alternating multilinear forms, differential forms on open subsets of ℝⁿ, the exterior derivative and pullback, integration over oriented chains, the general Stokes theorem, and closed versus exact forms leading to de Rham cohomology.

math.geom.differential-forms

Open lesson Show everything needed to reach this topic

Builds on

Multivariable and vector calculus Advanced linear algebra Real analysis

Required prerequisites, all the way down: 40

You will learn to

Compute with the wedge product, the exterior derivative and pullbacks, and verify d ∘ d = 0.
State the general Stokes theorem and recover the fundamental theorem of calculus, Green's, Stokes' and the divergence theorems as special cases.
Show that dθ on the punctured plane is closed but not exact, and prove the Poincaré lemma on star-shaped domains.

Definition

A differential k-form on an open set U ⊆ ℝⁿ assigns to each point an alternating k-linear form on ℝⁿ, depending smoothly on the point.
The exterior derivative d maps k-forms to (k + 1)-forms; on functions it is the differential df.
A form ω is closed if dω = 0 and exact if ω = dη for some form η.

Theorem

d ∘ d = 0, so every exact form is closed.
Stokes: for a compact oriented k-manifold M with boundary and a (k − 1)-form ω, ∫M dω = ∫{∂M} ω.
Poincaré lemma: on a star-shaped open set every closed k-form with k ≥ 1 is exact.
The 1-form dθ = (x dy − y dx)/(x² + y²) on ℝ² ∖ {0} is closed but not exact; its integral around the unit circle is 2π.

Common misconceptions

Thinking dθ is exact because it is written 'd' of something; θ is not a continuous single-valued function on the punctured plane.
Believing the wedge product is commutative; for a k-form α and an l-form β, α ∧ β = (−1)^{kl} β ∧ α.

Leads to

Vector bundles and characteristic classes Symplectic geometry and topology

Optional discoveries

Algebraic topology De Rham's theorem identifies the cohomology built from closed and exact forms with singular cohomology with real coefficients: calculus detects holes.
The geometry of spacetime: special and general relativity Maxwell's equations become dF = 0 and d⋆F = J in the language of forms, valid in any coordinates and on curved spacetime.
← Vectors and coordinate geometry