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Interactive Explainers

Play with the theorems

Interactive explainers that let you move the mathematics: drag ε and watch δ answer, walk the topologist's sine curve, trap a subsequence by bisection, watch Newton's method fall into a cycle. 15 are free to play right here — the rest come with Trap Analysis.

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Video explainers

Narrated walk-throughs of the trickiest ideas, made for this site — no ads, no recommendations pulling you elsewhere. Free to watch whether or not the interactive below is unlocked.

Unit 3Ordinary Differential Equations › Linear ODE, Wronskian, variation of parameters, systems#

The Wronskian: all or nothing, by Abel's identityinteractivefree

Step 1 / 4Abel's identity

For y′′+p(x)y′+q(x)y=0y'' + p(x)y' + q(x)y = 0 with p,qp,q continuous on an interval II, the Wronskian of two solutions satisfies W′=−pWW' = -pW, so

W(x)=W(x0)exp⁡(−∫x0xp).W(x) = W(x_0)\exp\left(-\int_{x_0}^x p\right).

The exponential never vanishes.

Unit 3Partial Differential Equations › Classification and canonical forms#

Classifying a second-order PDE, and the type that changesinteractivefree

Step 1 / 4The discriminant

For Auxx+Buxy+Cuyy+⋯=0Au_{xx} + Bu_{xy} + Cu_{yy} + \cdots = 0, compute B2−4ACB^2-4AC:

  • <0<0 elliptic (Laplace)
  • =0=0 parabolic (heat)
  • >0>0 hyperbolic (wave)

Unit 3Numerical Analysis › Root finding: bisection, Newton–Raphson, fixed point, order of convergence#

Newton's method: racing — and cyclinginteractive

This visual is part of Trap Analysis

Slide the starting point: quadratic convergence one moment, an eternal 2-cycle the next. Local convergence, felt.

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Unit 3Ordinary Differential Equations › Stability and phase portraits#

The trace–determinant planeinteractive

This visual is part of Trap Analysis

Drag a point across the plane and watch the portrait change: saddle, node, spiral, centre — a map instead of a list of rules.

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Unit 3Partial Differential Equations › Laplace, heat and wave equations: separation of variables#

d'Alembert: one bump becoming two wavesinteractive

This visual is part of Trap Analysis

Watch the initial profile split into halves travelling at ±c, with the light cone marking where the string has not moved yet.

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Unit 3Ordinary Differential Equations › Existence–uniqueness, Picard, Lipschitz#

Picard iteration converginginteractive

This visual is part of Trap Analysis

Start from a constant and watch successive integrals close on the solution — agreement spreading outward, which is why existence is only local.

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Unit 3Partial Differential Equations › First-order PDE: Lagrange, Charpit, characteristics#

Lagrange's method on a first-order PDEinteractive

This visual is part of Trap Analysis

The auxiliary system, two first integrals, and the general solution — worked end to end.

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Unit 3Ordinary Differential Equations › Sturm–Liouville problems and Green's functions#

Sturm–Liouville: what self-adjointness buysinteractive

This visual is part of Trap Analysis

Real eigenvalues, orthogonal eigenfunctions and simplicity — from one integration by parts.

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Unit 3Numerical Analysis › Numerical ODE: Euler, Runge–Kutta#

Order of a method, and the step size stability forces on youinteractive

This visual is part of Trap Analysis

Why explicit Euler needs h < 2/|λ| however accurate you wanted to be.

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Unit 3Numerical Analysis › Interpolation and numerical integration with error terms#

Degree of exactness, and why more nodes can be worseinteractive

This visual is part of Trap Analysis

Trapezium versus Simpson, and Runge's phenomenon in one example.

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Unit 3Calculus of Variations › Euler–Lagrange equation and standard functionals#

Euler–Lagrange, and the two shortcuts worth knowinginteractive

This visual is part of Trap Analysis

When F has no y, and when F has no x — each collapses the equation by one order.

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Unit 3Calculus of Variations › Isoperimetric problems#

Constrained extremals: the multiplier methodinteractive

This visual is part of Trap Analysis

Fixed perimeter, maximum area — set up with a Lagrange multiplier and solved.

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Unit 3Linear Integral Equations › Fredholm and Volterra equations#

Volterra always solvable, Fredholm not — and whyinteractive

This visual is part of Trap Analysis

The factorial in the iterated kernel that makes one type unconditionally well behaved.

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Unit 3Linear Integral Equations › Separable kernels and resolvent kernels#

Separable kernels turn an integral equation into linear algebrainteractive

This visual is part of Trap Analysis

A degenerate kernel reduces the whole problem to a finite system — worked.

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Unit 3Classical Mechanics › Lagrangian formalism and generalised coordinates#

Cyclic coordinates and the conservation laws they hand youinteractive

This visual is part of Trap Analysis

Central force in polar coordinates — angular momentum falls out of the Lagrangian in one line.

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Unit 3Classical Mechanics › Hamiltonian formalism and conservation laws#

From Lagrangian to Hamiltonian, and what the flow preservesinteractive

This visual is part of Trap Analysis

The Legendre transform done once, plus the reason a Hamiltonian system can never have an attractor.

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Every interactive explainer comes with Trap Analysis — along with the worked solutions, the per-option trap diagnosis and the counterexample objects. The papers, the book and the video map are free without it.

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