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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 1Continuity and Differentiation › Continuity, uniform continuity, Lipschitz#

The ε–δ limit machineinteractivefree

Unit 1Sequences and Series of Functions › Pointwise vs uniform convergence, M-test, Dini#

Watch uniform convergence fail: xⁿ on [0, 1]interactivefree

Unit 1Integration › Riemann integration and criteria#

Riemann sums closing on the integralinteractivefree

Unit 1Metric Spaces › Open/closed sets, limit points, closure, interior#

The unit ball, from diamond to squareinteractivefree

Unit 1Functions of Several Variables › Partial derivatives, differentiability, chain rule#

Partial derivatives can exist while the function is not differentiableinteractivefree

Step 1 / 4The candidate

Let

f(x,y)=xyx2+y2 for (x,y)≠(0,0),f(0,0)=0.f(x,y) = \frac{xy}{x^2+y^2} \text{ for } (x,y) \neq (0,0), \qquad f(0,0)=0.

Is ff differentiable at the origin?

Unit 1Metric Spaces › Compactness: open covers, sequential, Heine–Borel#

Why (0,1) is not compact — an explicit cover with no finite subcoverinteractivefree

Step 1 / 4The claim

(0,1)(0,1) is bounded, so if compactness were only about boundedness it would qualify. Show directly that it is not compact by exhibiting an open cover with no finite subcover.

Unit 1The Real Line › Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy#

Bolzano–Weierstrass: the bisection huntinteractive

This visual is part of Trap Analysis

sin n never converges — bisect its range and watch nested intervals trap a convergent subsequence anyway.

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Unit 1Inner Product Spaces and Forms › Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem#

Gram–Schmidt, one projection at a timeinteractive

This visual is part of Trap Analysis

Steer v₂ and watch its shadow on e₁ get subtracted — and see the algorithm genuinely halt when the vectors go dependent.

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Unit 1Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton#

What a matrix does to a vectorinteractive

This visual is part of Trap Analysis

Sweep a direction and watch Av swing — except on the eigendirections, where it only stretches. Det appears as the area of the image.

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Unit 1Sequences and Series of Functions › Power series, radius of convergence, Abel's theorem#

Taylor polynomials and the radius you can seeinteractive

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1/(1+x²) is smooth on all of ℝ, yet its series refuses to go past |x| = 1. Watch the wall the poles at ±i put there.

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Unit 1Vector Spaces and Linear Maps › Bases, dimension, rank–nullity#

Rank–nullity, used the way the exam uses itinteractive

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Three questions answered from one identity — including the bound most candidates get backwards.

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Unit 1Continuity and Differentiation › Differentiability, mean value theorems, Taylor, L'Hôpital#

Where L'Hôpital's rule quietly failsinteractive

This visual is part of Trap Analysis

A limit that exists but which L'Hôpital never finds — and the hypothesis everyone forgets to check.

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Unit 1The Real Line › Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements#

Choosing a convergence test without wasting timeinteractive

This visual is part of Trap Analysis

Which test to reach for, in what order, and the family where ratio and root both return 1.

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Unit 1The Real Line › limsup, liminf and subsequential limits#

Computing limsup and liminf without guessinginteractive

This visual is part of Trap Analysis

A worked sequence, the subsequential limit set, and the inequality that is only ever one-way.

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Unit 1The Real Line › Completeness, sup/inf, Archimedean property#

What completeness of ℝ actually buys youinteractive

This visual is part of Trap Analysis

The one axiom that separates ℝ from ℚ, and the three theorems that collapse without it.

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Unit 1Integration › Improper integrals and convergence tests#

The two exponents that decide most improper integralsinteractive

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Why 1/x^p flips at p = 1 in opposite directions at 0 and at ∞ — and the integral that converges but not absolutely.

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Unit 1Lebesgue Measure and Integration › Lebesgue integral, MCT, DCT, Fatou#

MCT, DCT and Fatou on one exampleinteractive

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The same escaping-mass sequence run past all three theorems — which apply, which do not, and why.

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Unit 1Lebesgue Measure and Integration › Measurable sets and functions#

A set with no measure: the Vitali constructioninteractive

This visual is part of Trap Analysis

Why Lebesgue measure cannot be defined on every subset of ℝ — built in four steps.

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Unit 1Lebesgue Measure and Integration › L^p spaces essentials#

Which Lᵖ contains which — and why it depends on the measureinteractive

This visual is part of Trap Analysis

The inclusion reverses between a probability space and the real line. Both directions, with the counterexample for each.

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Unit 1Metric Spaces › Completeness and Baire category#

Baire category, and the argument it powersinteractive

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Why ℝ is not a countable union of nowhere dense sets, and how that one fact proves several others.

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Unit 1Sequences and Series of Functions › Arzelà–Ascoli and equicontinuity#

Checking Arzelà–Ascoli: the hypothesis that usually failsinteractive

This visual is part of Trap Analysis

Two families that look harmless and are not equicontinuous — and exactly where each breaks.

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Unit 1Metric Spaces › Connectedness and path-connectedness#

Connected but not path-connectedinteractive

This visual is part of Trap Analysis

The topologist's sine curve, and why closure preserves connectedness but not path-connectedness.

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Unit 1Determinants and Matrix Tricks › Determinants, trace, block matrices, rank inequalities#

Block matrices: which determinant formulas are actually trueinteractive

This visual is part of Trap Analysis

The triangular block rule, and the tempting formula that fails unless the blocks commute.

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Unit 1Eigenvalues and Canonical Forms › Diagonalisability criteria#

Is this matrix diagonalisable? The decision procedureinteractive

This visual is part of Trap Analysis

The criterion that always settles it, and the two conditions that look sufficient but are not.

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Unit 1Eigenvalues and Canonical Forms › Jordan canonical form#

Building a Jordan form from the data you are giveninteractive

This visual is part of Trap Analysis

How block sizes are forced by the minimal polynomial and the kernel dimensions.

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Unit 1Eigenvalues and Canonical Forms › Rational canonical form#

Rational canonical form, and when to prefer itinteractive

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The form that exists over every field — and the divisibility chain that makes it unique.

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Unit 1Vector Spaces and Linear Maps › Linear transformations, matrix representation, change of basis#

Change of basis without sign errorsinteractive

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Which direction the change-of-basis matrix goes, and the similarity relation that follows.

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Unit 1Inner Product Spaces and Forms › Quadratic forms, positive definiteness, Sylvester's law#

Testing definiteness, and reading Sylvester's lawinteractive

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The leading-minors test done right — including the sign pattern for negative definite that catches people out.

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Unit 1Functions of Several Variables › Inverse and implicit function theorems, extrema#

Implicit function theorem: what to differentiate, and when it says nothinginteractive

This visual is part of Trap Analysis

A worked implicit derivative, plus the point where the hypothesis fails and the curve really does misbehave.

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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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