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 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 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.

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.

See pricing