Mathematics and mechanics · Print and keep
Reference sheets
I write these as dense single-subject sheets, meant to sit open beside you while you work: the identities, the transformations that generate them, and the traps, with nothing in between.
I typeset each one in LaTeX and size it for paper rather than a screen. I left the teaching out of them deliberately: the explanatory material lives in the guides and the longer book editions. I publish most sheets in English and Japanese.
The sheets
Nine · English & 日本語
Circle Geometry for the HKDSE
A three-page sheet for the circle geometry in the HKDSE Compulsory Part: chords, angles at the centre and in a segment, cyclic quadrilaterals, tangents and the alternate segment, the four centres of a triangle and circles on the coordinate plane. Every theorem has its diagram, its converse and the HKEAA reason abbreviation to write beside it, and the last page indexes every reason, theorem against converse.
PDFEnglish
HKDSE 円の幾何 要点シート
HKDSE 必修部分の円の幾何を 3 ページにまとめた。弦、中心角と円周角、円に内接する四角形、接線と接弦定理、三角形の外心・内心・重心・垂心、座標平面上の円を扱う。各定理に図と逆を添え、答案に書く HKEAA 式の理由の略記は英語のまま残した。
PDF日本語
Circle Geometry: both editions in one file
The English and Japanese editions bound together, English first, for printing or keeping as a single document.
PDFEnglish / 日本語
Olympiad Problem Solving: a Mini-Textbook for the IMO & Putnam
A 26-page handbook organised around what a problem looks like and which tool it is asking for: strategy and heuristics, inequalities, polynomials and functional equations, number theory, combinatorics, geometry and the Putnam's analysis and algebra. It carries fifty-seven worked examples, most of them from IMO and Putnam papers, a formula compendium and a problem ladder by technique.
PDFEnglish
数学オリンピックの問題解決：IMO・Putnam のためのミニ教科書
問題の姿からどの道具が求められているかを読み取る、という軸で組んだ 28 ページの手引き。戦略と発見法、不等式、多項式と関数方程式、整数論、組合せ論、幾何、Putnam の解析と代数を扱い、IMO と Putnam の過去問を中心とする例題 57 題、公式集、技法別の演習リストを収めた。
PDF日本語
Olympiad Problem Solving: both editions in one file
The English and Japanese editions bound together, English first, for printing or keeping as a single document.
PDFEnglish / 日本語
Problem-Solving Methods: Calculus to Vector Calculus
Forty-two blocks of methods, key formulas and fully worked examples with figures, from limits and integration technique through polynomials, trigonometry, complex numbers, matrices and counting to partial derivatives, Lagrange multipliers, multiple integrals and the theorems of Green, Stokes and Gauss. It ends with 88 practice problems and their answers.
PDFEnglish
解法の手引き：微積分からベクトル解析まで
極限と積分の技法から、多項式、三角法、複素数、行列、数え上げを経て、偏微分、ラグランジュの未定乗数法、重積分、グリーン・ストークス・ガウスの定理まで。42 のブロックに解法、主要公式、図入りの例題を収め、巻末に解答つきの演習問題 88 問を置いた。
PDF日本語
Problem-Solving Methods: both editions in one file
The English and Japanese editions bound together, English first, for printing or keeping as a single document.
PDFEnglish / 日本語
Combinatorics: Permutations, Combinations & Identities
Sampling with and without replacement, the core binomial identities, hockey stick and Vandermonde, stars and bars, inclusion–exclusion and derangements, the twelvefold way, Stirling, Bell and Catalan numbers, Lucas and Kummer, and entropy bounds.
PDFEnglish
組合せ論：順列・組合せと二項係数の恒等式
標本抽出の四類型、二項係数の主要な恒等式、ホッケースティックとヴァンデルモンドの畳み込み、仕切り棒法、包除原理と完全順列、十二通りの道、スターリング数・ベル数・カタラン数、リュカとクンマーの定理、そしてエントロピーによる評価。
PDF日本語
Products & ∏ Notation
Telescoping, log–exp duality, falling and rising powers, Gamma and Gauss multiplication, roots of unity, q-products, convergence tests for infinite products, and Weierstrass–Hadamard factorisation.
PDFEnglish
積の要約：∏ の恒等式と変形
テレスコープ、対数と指数の双対性、下降べきと上昇べき、ガンマ関数とガウスの乗法公式、1 の冪根、q-積、無限積の収束判定、そしてワイエルシュトラス・アダマールの因数分解。
PDF日本語
Taylor Expansions: Single & Multivariable
A four-page sheet built around pictures: partial sums closing in on sin x, error slopes on log–log axes, why 1/(1+x²) stops converging at |x| = 1, and level curves settling onto their quadratic models at a minimum and a saddle. Eleven worked examples cover the series toolkit, error bounds, limits, gradients and Hessians, and the second-derivative test, with a self-test at the end.
PDFEnglish
Quaternions
A three-page sheet built around pictures: rotation as the sandwich q v q* and why the angle is halved, the double cover, conversions to and from matrices and Euler angles (with Shepperd’s method), slerp against nlerp, kinematics and gimbal lock, the conventions that differ between libraries, and recipes such as swing–twist and orientation averaging. The interactive topic accompanies it.
PDFEnglish
四元数
サンドイッチ q v q* としての回転と半角の理由、二重被覆、行列・オイラー角との変換、slerp と nlerp、運動学とジンバルロック、ライブラリごとの流儀の違い、よく使う手順。
PDF日本語
Push-forward and pullback
A three-page sheet on how integrals travel along a map: what pulls back and what pushes forward, pulling back forms and the Jacobians to know, the signed integral and degree, the area formula with multiplicity, push-forward measures and their densities (with the fold singularities), the coarea formula and fibre integration, flows, transport maps, and the conventions that bite. The interactive encyclopedia topic accompanies it.
PDFEnglish
押し出しと引き戻し
写像に沿った積分の移り方。微分形式の引き戻しとヤコビアン、符号つきの積分と次数、重複度つきの面積公式、押し出し測度とその密度、余面積公式、流れと輸送、食い違いやすい流儀。
PDF日本語
Mechanical Reasoning: a Field Guide
A four-page sheet for mechanical-reasoning problems, built around one idea: every machine trades force for distance. Pulleys, simple and compound gear trains, levers, hydraulic pistons and water pressure, centre of gravity and tipping, and beams on two supports each get a diagram, the governing law, a worked example checked by a second route, and the mistakes the wrong options are built from. It ends with the wider test topics, a table of trap patterns and an eight-problem drill.
PDFEnglish
機械的推論：実戦ガイド
「機械はすべて力と距離の交換である」という一つの原理を軸に組んだ 4 ページのシート。滑車、単純歯車列と複合歯車、てこ、油圧ピストンと水圧、重心と転倒、二点で支えたはりのそれぞれに、図、支配する法則、別の道筋で検算した例題、誤答の選択肢の元になる誤りを添えた。
PDF日本語
Mechanical Reasoning: both editions in one file
The English and Japanese editions bound together, English first, for printing or keeping as a single document.
PDFEnglish / 日本語
Laurent and Bessel Visual Lab
Explore annuli, partial sums, Bessel waves, and the series geometry behind the reference sheet.
InteractiveEnglish / 日本語
How they are made
A sheet earns its place by what it leaves out.
I compile everything here from LaTeX sources: Palatino for the mathematics, Noto Serif CJK for the Japanese editions. I lay the sheets out in two columns on A4 and check that they print cleanly in black and white, which is the only test I trust for a page you intend to keep beside you.