Notes

Calculations, diagrams, and proofs

Expository writing and visual material developed while studying geometry, measure theory, analysis, and the history of mathematics.

K = det(Hess f) / (1 + |∇f|²)²

Differential geometry · July 2026

The Gaussian Curvature Formula for Graphs of Functions on R²

For a graph Σ of a C² function, the note computes Gaussian curvature by taking the determinant of the derivative of the upward unit normal. Coordinate curves provide a tangent basis, while projection to the xy-plane makes the matrix coefficients transparent.

At a critical point of the function, the denominator becomes one, and the curvature is exactly the determinant of the Hessian.

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Premeasure → Outer measure → Measure

Measure theory · 2024 · 1 page

The Construction of Abstract Measures

A one-page structural diagram showing the passage from a premeasure to outer measures, Carathéodory measurability, completion, and the resulting measure spaces.

The diagram also marks where σ-finiteness gives uniqueness or simplifies the construction.

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L^q ⊂ L^p or L^p ⊂ L^q

Functional analysis · 2024 · 2 pages

Monotonicity of Lebesgue Spaces

With Zixuan Wang. This note characterizes when the inclusions L^q ⊂ L^p and L^p ⊂ L^q hold on a general measure space.

The conditions are expressed through upper and lower bounds on the measures of nontrivial measurable sets.

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∞

Lecture slides · December 2025 · 19 slides

The History of Infinity

A historical overview of infinity, from early philosophical and mathematical debates to its role in modern logic and set theory.

The slides were prepared for an online mathematics presentation and emphasize the changing meanings of potential and actual infinity.

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About the collection

This page gathers polished expository notes, diagrams, and lecture materials rather than a complete archive of working calculations. The emphasis is on self-contained arguments and visual organization that may be useful to students.