Library

The flagship application of the integral: measuring the area of a region of the plane. From the signed area under a curve to the area between two curves, without getting the sign wrong.

Why an ambulance siren changes pitch as it passes by, and how the same effect, applied to the light of galaxies, revealed the expansion of the Universe.

How heat moves: through contact (conduction), through the motion of a fluid (convection) or across a vacuum (radiation). The laws that govern insulation and the heat balance.

The force that governs attraction and repulsion between electric charges: a 1/r² law, twin sister of gravitation, but billions of billions of times more intense.

A single equation, PV = nRT, links the pressure, volume, temperature and amount of a gas. Where it comes from, what it hides at the scale of molecules, and its limits.

Where does the integral come from? From the very simple idea of approximating an area with thinner and thinner rectangles. The definition that links discrete sums to integral calculus.

The key technique for integrating composite functions: spot a hidden derivative and "rename" the variable. The reverse of the chain rule.

How do you average infinitely many values? The integral answers: the average value of a function on an interval, its geometric meaning, and the mean value theorem for integrals.

When no antiderivative exists, we approximate the integral by computation. From rectangles to trapezoids and then to Simpson's rule: more and more accurate, with the same number of points.

A complete take-home assignment: bound ln(1 + x) by integration, study the function f(x) = ln(1 + x)/x, then deduce the convergence speed of the sequence u(n+1) = ln(1 + u(n)).

Forty-one rebels in a circle, eliminated three by three: where should you stand to survive? A take-home assignment leading from a Python simulation to an exact formula readable off the binary expansion.

How a moving magnet gives birth to a current, without a battery. Faraday's law and Lenz's law — and how they produce nearly all of our electricity.

A single number that decides everything: how the determinant measures an area or a volume, how to compute it without exhausting yourself, and why it vanishes exactly when a matrix fails to be invertible.

Writing a system as AX = B and then solving it: what the inverse of a matrix is, how to compute it by Gauss-Jordan, the ten equivalent ways of saying a matrix is invertible, and why elimination is still preferred.

This course explores the major principles of thermodynamics, from the notion of temperature to the efficiency of heat engines. You will discover how energy is conserved and degraded, and what this imposes on engines and refrigerators.

Counting infinity: why the rationals can be numbered but the reals cannot, and how Cantor's diagonal runs through logic as well as theoretical computer science.

RSA is solid on paper, but deadly if misused. Why "textbook" RSA is dangerous, and the catalogue of classic attacks: determinism, small exponent, bad parameters and padding oracles.

What if we could compute on data without ever decrypting it? From theoretical dream to fully homomorphic encryption, the cryptography that processes without seeing.

How can several parties jointly compute a result without ever revealing their private data? From Yao's millionaires' problem to garbled circuits and the real-world uses of MPC.

How does a messaging app protect past messages even if your phone is stolen? A deep dive into forward secrecy and Signal's double ratchet algorithm.

How do you prove your identity beyond the password? From six-digit codes to cryptographic keys that make phishing impossible.

How to prove you know a secret key without ever showing it, and how this interactive proof becomes one of the most elegant signatures in modern cryptography.

Cryptography makes a message unreadable; steganography hides the very fact that a message exists. How to hide a secret in an image, how one tries to detect it, and why you should always encrypt before hiding.

Discover how a wave carries energy without carrying matter, and understand the phenomena that govern sound: interference, standing waves, the Doppler effect and beats.