Glossary
Each lesson opens with a “Before you begin” list of what it assumes you already know. Every item links to its entry here. An entry is a reminder, not a course: it says what the lessons mean by the term. Where a lesson on this site explains the topic, the entry links to it.
Algebra
Manipulating expressions with symbols: expanding products, factorising, and collecting terms until two expressions can be compared.
Assumed in: Proving what the simulation showed, What symplectic Euler conserves, Lean exercises on your machine.
Calculus
The derivative of a function is its rate of change, and the integral accumulates a rate over an interval. Taylor expansion approximates a smooth function near a point by a polynomial in the distance from it.
Assumed in: Kinematics as a program, Newton’s laws as a first-order system, The harmonic oscillator, Numerical integrators, Energy conservation, Double pendulum and chaos, Lagrangian mechanics, Hamiltonian mechanics, Resonance and driven oscillators, Kinetic theory of gases, The wave equation, Sound, resonance, and the Doppler effect, Electric fields, superposition, and the potential, Induction: magnetic flux and the electromotive force, Advection and diffusion: schemes, stability, and the diffusion a grid adds, Shock waves and the Burgers equation, Finite-volume methods and conservation laws, Shallow-water equations: still water and the dam break, Discovering differential equations from trajectories, The chirp law, proved.
Complex numbers
A complex number is a point in a plane, written with a real and an imaginary part. Its modulus is the distance from the origin, its argument the angle, and the exponential of an imaginary number is a point on the unit circle.
Assumed in: Numerical integrators, Gravitational-wave strain lab, Resonance and driven oscillators, Interference and resonance, Advection and diffusion: schemes, stability, and the diffusion a grid adds.
Geometry
The shapes the lessons use. An ellipse is the set of points whose distances to two foci add up to a constant; its semi-major axis is half its longest width and its eccentricity says how far it is from a circle.
Assumed in: The Kepler orbit, Hamiltonian mechanics.
Large language models
A large language model is a program that, given a text, continues it with the words that are likely to follow. It has no access to anything but the text it is given, unless a program hands it tools.
Assumed in: An agent runs an experiment, A counterexample hunter, Discovering a force law with controlled probes.
Linear algebra
A matrix maps a vector to a vector; applying it again and again is how a linear step rule advances a state. The determinant of a two-by-two matrix is the factor by which it scales areas, its trace is the sum of its diagonal, and its eigenvalues are the factors by which it stretches the directions it leaves unturned.
Assumed in: Numerical integrators, Double pendulum and chaos, Lagrangian mechanics, Hamiltonian mechanics, Discovering differential equations from trajectories.
Logarithms
The logarithm to a base is the power that base is raised to in order to give a number: the base-2 logarithm of 8 is 3. On a log-log plot both axes are logarithmic, so a power law shows as a straight line whose slope is the exponent.
Assumed in: Kinematics as a program, Projectile motion with quadratic drag, The harmonic oscillator, Double pendulum and chaos, Entropy from ensembles, Discovering a force law with controlled probes, Unlearning a familiar law: a modified force, Symbolic regression: a law from simulated data.
Logic
A statement that says if these hypotheses hold, then this conclusion holds says nothing when a hypothesis fails. A proof derives the conclusion from the hypotheses by steps that each follow from the rules.
Assumed in: Proving what the simulation showed, Lean exercises on your machine.
Numerical methods
Methods that compute an approximate answer by a finite sequence of arithmetic steps. The Newton iteration finds a root of an equation by replacing the function with its tangent line and repeating from the root of that line.
Explained in Numerical integrators.
Assumed in: The Kepler orbit, The wave equation.
Probability
A probability distribution says how likely each outcome is. The Gaussian distribution is the bell curve set by a mean and a standard deviation, and the t distribution is a wider bell curve for a mean estimated from few samples.
Assumed in: Random walks and diffusion, Kinetic theory of gases, Entropy from ensembles, Particles in a box.
Python
The programming language of the lessons. You should be able to read and write functions, dictionaries, and dataclasses, to use NumPy arrays, and to read environment variables.
Assumed in: Kinematics as a program, Verifying scientific code, An agent runs an experiment, A counterexample hunter, Three reviewers for one piece of code, An agent that designs the measurements.
Statistics
The mean of samples estimates a quantity, and the standard deviation says how far single samples scatter around it. The standard error of the mean is the standard deviation divided by the square root of the number of samples, and says how far the mean itself is uncertain.
Assumed in: Automated analysis pipelines, Gravitational-wave strain lab, Inverse physics: parameters from observations, Random walks and diffusion, Kinetic theory of gases, Entropy from ensembles, An agent that designs the measurements, Full, partial, and no prior knowledge under one protocol, Unlearning a familiar law: a modified force, Symbolic regression: a law from simulated data, Equations from noisy trajectories: where differentiation fails, Particles in a box.
Trigonometry
A vector of length r at angle theta to the x axis has the components r cos(theta) and r sin(theta). For small theta the sine of theta is close to theta itself, with the angle in radians.
Assumed in: Projectile motion with quadratic drag, Energy conservation, Double pendulum and chaos, Lagrangian mechanics, Resonance and driven oscillators, Interference and resonance, Charged particles in electric and magnetic fields, Full, partial, and no prior knowledge under one protocol.
Vectors
A vector is a quantity with a direction, written as components along the axes. Vectors add and scale component by component; the dot product of two vectors is a number, the cross product in a plane is a number for the area they span, and the gradient of a function of the position is the vector of its derivatives along each axis. In three dimensions the cross product is a vector perpendicular to both, and a × (b × c) = b (a·c) − c (a·b).
Assumed in: Newton’s laws as a first-order system, Energy conservation, Momentum and collisions, The Kepler orbit, Electric fields, superposition, and the potential, Charged particles in electric and magnetic fields, Induction: magnetic flux and the electromotive force, Momentum in a collision, proved.