Reading a graph
A graph plots dependent outputs against independent inputs on two axes. The slope measures the rate of change and the area under a curve represents an accumulated sum.
A finite path through the prerequisites
Start with the idea
that is missing.
Each lesson includes a worked example and a stopping point. Follow the prerequisites as far as you need, then return to the argument.
Newly authored explanatory previews; editorial review remains pending.
Read the Brownian argument →Fractions and ratios
A fraction compares a part to a whole or one quantity to another. Scaling numerator and denominator by the same factor preserves the ratio.
A sign records direction
Choose right as positive and left as negative. Opposite displacements can cancel without either particle staying still.
Powers of ten and physical units
Powers of ten shift the decimal place. A physical quantity is a number multiplied by its unit; dropping the unit destroys its physical meaning.
Squares and square roots
Squaring multiplies a number by itself. The nonnegative square root reverses that operation.
Adding and averaging
Add the values, then divide by how many there are. An average does not preserve the individual values.
Rates of change and derivatives
A derivative measures the instantaneous rate of change of one physical quantity with respect to another, defined as the limit of average differences over shrinking intervals.
What the diffusion coefficient means
D measures the rate of growth of a coordinate’s mean square: on an unbounded line that growth is 2D per unit time.
Density is not probability
A density measures probability per unit of position. Probability belongs to an interval and is its area under the curve.
Forces on charges, currents, and electromagnetic waves
Electric charges produce electric fields and experience forces q E; moving charges experience magnetic forces q v x B. Bound charges act as resonators absorbing and emitting electromagnetic waves at the speed of light.
Entropy, temperature, and a stated constraint
At fixed volume and other stated constraints, the equilibrium entropy derivative with respect to energy is inverse absolute temperature. The derivative does not determine an additive constant.
Uncertainty, evidence and inference
Keep identifiability, estimator bias, repeated-sample uncertainty and independently measured inputs separate.
Exponentials and continuous scaling
The exponential function describes systems where the rate of change is proportional to current value, requiring a strictly dimensionless exponent.
Fields, continuous waves, and harmonic functions
A field assigns physical values across continuous space. Harmonic waves propagate oscillations with phase kx - omega t, where optical intensity reflects the time average of cos^2 equal to 0.5 rather than instantaneous fluctuations.
Counting what crosses a boundary
What accumulates in a region equals what enters minus what leaves, provided nothing is created or destroyed inside.
Events, reference frames, and the coordinate grid
An event is a physical occurrence at a single point in space and instant in time. A reference frame is an idealized coordinate system of rigid rods and synchronized clocks that assigns four numbers (x, y, z, t) to each event.
Functions and graphs
A function pairs each input value with exactly one output value; its graph displays that pairing as a continuous curve where coordinates and slopes carry physical units.
The Gaussian and its width
A Gaussian is a family of densities whose spread is set by its variance; being bell-shaped is not a proof of its origin.
Adding continuously
An integral adds contributions from narrow intervals; its units include the width of each interval.
Logarithms and product-to-sum relations
A logarithm is the inverse of exponentiation, converting state-count multiplications into additive entropy, and carrying historical notation distinctions.
Mean, variance and RMS
The mean tracks the centre; the variance tracks spread around the centre; RMS measures distance from the chosen zero.
Momentum, energy flux, and radiation pressure of light
Light carries momentum proportional to its energy: p = E / c. Absorbing 1 W of light produces a continuous push of 1/c = 3.3356e-9 N, while ideal reflection doubles the push.
Partial derivatives and held-fixed quantities
A partial derivative quantifies how a multivariable function responds to variations in one variable while all other independent variables are strictly held constant.
Probability and independence
Independent centred steps have zero expected product. Independence alone is not enough: their means must also be zero.
From steps to spread
Independent centred increments add their variances, not their distances.
A local expansion
Near a point, value, slope and curvature describe the leading change. Discarding the remaining terms requires a scale argument.
Energy of motion and inertia
At low speeds, energy of motion depends on both speed and inertia. Keeping speed fixed lets us compare the inertia before and after a change.