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.

A set with no measure: the Vitali construction
Lebesgue Measure and Integration › Measurable sets and functions
Baire category, and the argument it powers
Metric Spaces › Completeness and Baire category

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 2Zeros and Mappings › Argument principle, Rouché's theorem, open mapping#

Rouché in action: counting the zeros of z⁵ + 3z + 1interactivefree

Step 1 / 5The question

How many zeros does $p(z) = z^5 + 3z + 1$ have inside $|z| < 1$ ? And inside $|z| < 2$ ?

No formula finds these roots — but Rouché's theorem counts them without finding them.

Unit 1Integration › Riemann integration and criteria#

Riemann sums closing on the integralinteractivefree

Unit 4Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT#

The Central Limit Theorem, watchedinteractivefree

Unit 4Hypothesis Testing › Neyman–Pearson lemma and UMP tests#

Type I against Type II: the trade-off you cannot escapeinteractivefree

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

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

Is $f$ 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

$(0,1)$ 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 2Analytic Functions › Cauchy–Riemann equations, harmonic functions#

Cauchy–Riemann: satisfied at a point, holomorphic nowhereinteractivefree

Step 1 / 4The equations

Writing $f = u+iv$ , complex differentiability at a point requires

$u_x = v_y, \qquad u_y = -v_x.$

These are necessary. They are not, on their own, sufficient.

Unit 2Cauchy Theory › Cauchy's theorem and integral formula#

Cauchy's integral formula, and what it gives you for freeinteractivefree

Step 1 / 4The theorem

If $f$ is holomorphic on and inside a simple closed contour $C$ , then $\oint_C f = 0$ . If $a$ is inside $C$ ,

$f(a) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a}\,dz.$

The value at an interior point is determined entirely by the boundary values.

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 $y'' + p(x)y' + q(x)y = 0$ with $p,q$ continuous on an interval $I$ , the Wronskian of two solutions satisfies $W' = -pW$ , so

$W(x) = W(x_0)\exp\left(-\int_{x_0}^x p\right).$

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 $Au_{xx} + Bu_{xy} + Cu_{yy} + \cdots = 0$ , compute $B^2-4AC$ :

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

Unit 4Probability › Axioms, conditional probability, independence, Bayes#

Bayes and the base rate: why a 99% accurate test is usually wronginteractivefree

Step 1 / 4The setup

A test has sensitivity $0.99$ and specificity $0.95$ ; the disease affects $1\%$ of people. You test positive. What is the chance you have it?

Unit 4Probability › Standard discrete and continuous distributions#

A distribution's own shape, as its parameters moveinteractivefree

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.

Play with the theorems

Video explainers

The ε–δ limit machineinteractivefree

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

Rouché in action: counting the zeros of z⁵ + 3z + 1interactivefree

Riemann sums closing on the integralinteractivefree

The Central Limit Theorem, watchedinteractivefree

Type I against Type II: the trade-off you cannot escapeinteractivefree

The unit ball, from diamond to squareinteractivefree

Partial derivatives can exist while the function is not differentiableinteractivefree

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

Cauchy–Riemann: satisfied at a point, holomorphic nowhereinteractivefree

Cauchy's integral formula, and what it gives you for freeinteractivefree

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

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

Bayes and the base rate: why a 99% accurate test is usually wronginteractivefree

A distribution's own shape, as its parameters moveinteractivefree

Bolzano–Weierstrass: the bisection huntinteractive

The topologist's sine curve, walkedinteractive

Gram–Schmidt, one projection at a timeinteractive

Why periodic chains never settleinteractive

Newton's method: racing — and cyclinginteractive

What a matrix does to a vectorinteractive

The trace–determinant planeinteractive

Taylor polynomials and the radius you can seeinteractive

Möbius maps bending the gridinteractive

What 95% confidence actually meansinteractive

d'Alembert: one bump becoming two wavesinteractive

Picard iteration converginginteractive

Lagrange's theorem, tiledinteractive

Rank–nullity, used the way the exam uses itinteractive

Where L'Hôpital's rule quietly failsinteractive

Choosing a convergence test without wasting timeinteractive

Computing limsup and liminf without guessinginteractive

What completeness of ℝ actually buys youinteractive

The two exponents that decide most improper integralsinteractive

MCT, DCT and Fatou on one exampleinteractive

A set with no measure: the Vitali constructioninteractive

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

Baire category, and the argument it powersinteractive

Checking Arzelà–Ascoli: the hypothesis that usually failsinteractive

Connected but not path-connectedinteractive

Block matrices: which determinant formulas are actually trueinteractive

Is this matrix diagonalisable? The decision procedureinteractive

Building a Jordan form from the data you are giveninteractive

Rational canonical form, and when to prefer itinteractive

Change of basis without sign errorsinteractive

Testing definiteness, and reading Sylvester's lawinteractive

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

Classifying an isolated singularity in one lookinteractive

Computing residues without expanding the seriesinteractive

Liouville, and the growth bounds that do not force constancyinteractive

Why the radius of convergence is set by the nearest singularityinteractive

Lagrange's theorem and its false converseinteractive

The first isomorphism theorem, used as a toolinteractive

Cycle type decides everything in Sₙinteractive

The class equation, and what it proves about p-groupsinteractive

Sylow counting: proving a group is not simpleinteractive

Classifying finite abelian groups, and telling two apartinteractive

Prime versus maximal, decided by the quotientinteractive

Euclidean ⊂ PID ⊂ UFD, with a separating example at each stepinteractive

Irreducibility tests, in the order worth trying theminteractive

Degrees multiply — computing [K:ℚ] and spotting the primitive elementinteractive

The Galois correspondence, used on one exampleinteractive

Which properties survive subspaces, products and quotientsinteractive

The compactness facts that need Hausdorff, and the ones that do notinteractive

The separation axioms, in the order they matterinteractive

The standard counterexample spaces, and what each is forinteractive

Lagrange's method on a first-order PDEinteractive

Sturm–Liouville: what self-adjointness buysinteractive

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

Degree of exactness, and why more nodes can be worseinteractive

Euler–Lagrange, and the two shortcuts worth knowinginteractive

Constrained extremals: the multiplier methodinteractive

Volterra always solvable, Fredholm not — and whyinteractive

Separable kernels turn an integral equation into linear algebrainteractive

Cyclic coordinates and the conservation laws they hand youinteractive

From Lagrangian to Hamiltonian, and what the flow preservesinteractive

Moments, MGFs, and when they fail to existinteractive

The signatures that identify a distribution instantlyinteractive