Mathematical Foundations is the essential toolkit for mastering mechanics. Rather than treating math as a set of isolated abstract rules, this track presents key mathematical concepts—from vector geometry to calculus—as the physical language nature uses to describe motion, force, and change. Each lesson builds intuitive, visual, and operational mastery so you can stop wrestling with equations and start using them to analyze the physical world.
Module 1: Mathematical Foundations · Practice · assumes Foundations 5 and 6 Five worked examples on the two vector products — projection, power, torque in three dimensions, and angular momentum. What this set is for The earlier worked examples treated the dot product only as a way of computing work, and the cross product only … Read more
Module 1: Mathematical Foundations · Practice · assumes Foundation 9 Six worked examples on integration — what it is, the two techniques worth knowing, and the identity that removes time from a problem. What this set is for Foundation 9 introduced substitution and integration by parts once each, and Foundation 11 called a dx = … Read more
Module 1: Mathematical Foundations · Practice · assumes Foundations 3 and 8 Six harder examples on one operation: differentiating when the basis vectors themselves are turning. What this set is for Foundation 8 derived the two results everything here rests on: dr̂/dt = θ̇θ̂ and dθ̂/dt = −θ̇r̂, and from them the velocity and acceleration … Read more
Module 1: Mathematical Foundations · Practice · complements Foundations 2, 3, 4 and 7 Seven examples on the one question this module keeps returning to: what belongs to the physics, and what belongs to the description. What this set is for The first worked-examples post took one problem from each Foundation and covered the standard … Read more
Module 1: Mathematical Foundations · Practice · complements Foundations 1–11 One worked example for each Foundation — the calculation in full, and the question a physicist would ask around it. How to use this page Every example is set out in three parts. The Physics asks what is happening, before any symbol is written. The … Read more
Module 1: Mathematical Foundations · Reference · consolidates Foundations 1–10 Everything from this module in one place — built for scanning, not for reading. What this module gave you Ten readings, and underneath them a single argument. Physics describes the world with mathematics, and every description carries a domain outside which it stops being true. … Read more
Module 1: Mathematical Foundations · Theory · assumes Foundation 8 The techniques that make hard problems solvable — and the discipline of knowing when they stop working. Mahavakya There is no exact treatment of anything. Every problem you have solved was an approximation — the skill is knowing which one you made. On this page … Read more
Module 1: Mathematical Foundations · Theory · assumes Foundations 7 and 8 Adding up what changes continuously — and discovering that this is differentiation run backwards. Mahavakya To add up something that never holds still, cut it into pieces so small that on each one it does. On this page Slicing, and the Riemann sum … Read more
Module 1: Mathematical Foundations · Theory · assumes Foundations 3 and 7 Computing what Foundation 7 taught you to see — the exact rate of change at a single instant. Mahavakya A derivative is a rate measured over no interval at all. Nothing changes in an instant — and yet the change at that instant … Read more
Module 1: Mathematical Foundations · Theory · assumes Foundation 1 Learning to read a curve — before learning to compute on one. Mahavakya A graph is not a picture of a formula. It is the whole behaviour of a system, laid out where the eye can take it in at once. On this page What … Read more
