Tag
Analysis
Analysis is the study of limits, and of the structures that survive them: continuity, differentiation, integration, and convergence. Its modern form treats the derivative as a linear map rather than a slope, which is what allows it to move off the line and onto manifolds. Writing here builds analysis in the direction of differential forms and Stokes' theorem, where the fundamental theorem of calculus turns out to have been one case of a statement about boundaries.
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.
April 12, 2026
Analysis on Manifolds II: Exterior Algebra
The algebraic machinery behind differential forms: dual spaces, multilinear alternating maps, the wedge product, bases and dimension of Λᵏ(V*), determinants as top forms, the interior product, and the Hodge star. Part II of a five-part lecture series on differential forms and the generalised Stokes theorem.
March 28, 2026
Analysis on Manifolds I: Analysis on ℝⁿ
A self-contained, proof-based reconstruction of single and multivariable analysis from first principles: topology of ℝⁿ, the derivative as a linear map, the chain rule, and the inverse and implicit function theorems. Part I of a five-part lecture series on differential forms and the generalised Stokes theorem.