Tag
Topology
Topology studies the properties of a space that survive continuous deformation: connectivity, compactness, the number of holes. It supplies the vocabulary in which convergence and continuity are stated without reference to distance, and the invariants that distinguish spaces no coordinate calculation can separate. Writing here meets topology through integration on manifolds, where orientation, boundary and compact support decide which integrals are defined.
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.