SJShaoda Ji

Notes

Calculations, diagrams, and proofs

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

\[\Delta_g r=\partial_r\log\sqrt{\det g}\]
Differential geometry · 6 pages

The Laplace–Beltrami Operator in Spherical and Geodesic Normal Coordinates

This note derives the Laplace–Beltrami operator first in ordinary spherical coordinates on \(\mathbb{R}^N\), then develops its analogue in Riemannian normal and geodesic polar coordinates.

The discussion isolates the role of the Riemannian volume density and shows how Ricci curvature enters the first correction to the Euclidean radial Laplacian near the center.

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\[\sqrt{\det\big((\nabla f)^T\nabla f\big)}\]
Geometric analysis · July 2026 · 6 pages

A Geometric Approach to Integration

This note introduces integration from a geometric point of view, beginning with familiar low-dimensional examples and using approximation and accumulation as the guiding principles of measurement.

It moves from derivatives and differentials to Gram matrices and Jacobians, then explains how these ideas assemble into the general parametric integral.

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\[K=\frac{\det(\operatorname{Hess} f)}{(1+\lvert\nabla f\rvert^2)^2}\]
Differential geometry · July 2026

The Gaussian Curvature Formula for Graphs of Functions on \(\mathbb{R}^2\)

For a graph \(\Sigma\) of a \(C^2\) 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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\[\mu^*(A)=\inf\sum_{n=1}^{\infty}\mu_0(E_n)\]
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 \(\sigma\)-finiteness gives uniqueness or simplifies the construction.

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\[L^q\subset L^p\;\text{or}\;L^p\subset L^q\]
Functional analysis · 2024 · 2 pages

Monotonicity of Lebesgue Spaces

With Zixuan Wang. This note characterizes when the inclusions \(L^q\subset L^p\) and \(L^p\subset L^q\) 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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\[\infty\]
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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\[\operatorname{Rot}(\mathbb{S}^2)\cong\operatorname{PSU}(2)\]
Complex analysis and geometry · December 2025 · 2 pages

Rotations on the Riemann Sphere

This note studies when a Möbius transformation induces a rotation of the Riemann sphere under stereographic projection.

Using Hermitian inner products and a linear-algebra reduction, it shows that the inducing matrix is a scalar multiple of a unitary matrix, giving the identification \(\operatorname{Rot}(\mathbb{S}^2)\cong\operatorname{PSU}(2)\).

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\[L(f)=\int_X f\,\mathrm{d}\mu\]
Measure theory · 2025 · 10 pages

Radon Measures and Riesz Representation Theorems

A seminar note developing Radon measures on locally compact Hausdorff spaces and the topological tools used in their construction.

It presents two Riesz representation theorems and concludes with a compactness theorem for locally bounded sequences of Radon measures.

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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.