Tag
Manifolds
A manifold is a space that looks locally like a vector space while being permitted to bend globally, which is the setting in which calculus continues to work once flatness is given up. Charts supply the local coordinates, transition functions record how those coordinates disagree, and the objects worth naming are the ones no chart can see. Writing here builds manifolds as the home of differential forms and integration, and keeps the distinction between an object and its coordinate description in the foreground.
Blog
April 14, 2026
Analysis on Manifolds V: Stokes’ Theorem
The generalised Stokes theorem proved in full, recovering the fundamental theorem of calculus, Green’s theorem, the divergence theorem, and the classical Stokes theorem as special cases. Hodge decomposition with complete Sobolev proof. Harmonic representatives, Betti numbers, and the de Rham isomorphism. The conclusion of a five-part lecture series.
April 13, 2026
Analysis on Manifolds IV: Integration
Forms as antiderivatives, oriented manifolds, manifolds with boundary, partitions of unity, integration of k-forms over k-submanifolds, the change-of-variables theorem via pullback, the Riemannian volume form, period integrals, the Mayer-Vietoris sequence, and the full de Rham cohomology of spheres, tori, and surfaces. Part IV of a five-part lecture series on differential forms and the generalised Stokes theorem.
April 13, 2026
Analysis on Manifolds III: Differential Forms
Smooth manifolds, tangent and cotangent spaces, differential forms as smooth sections of Λᵏ(T*M), the exterior derivative, pullback along smooth maps, and the recovery of grad, curl, and div as the exterior derivative in ℝ³. Part III of a five-part lecture series on differential forms and the generalised Stokes theorem.