Interactive visual explainers for CSIR NET Maths

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.

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

Unlock every visual

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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