# Chapter ᚉ(C): The gallery of motion *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T11, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [Its glossary cards](ch11-cards.md) · [Its references](ch11-refs.md) · [Previous chapter](ch10.md) · [Next chapter](ch12.md) *A footnote such as [r-11.3] is reference entry 11.3 and links to it. Section 11 of 28; chapters are named by their Ogham letter, never by number.* ## The five-room museum ᚉ – [Coll](ch11-cards.md#g-og-coll): associated with the hazel tree, Section 11 ## Chapter ᚉ(C): The gallery of motion > “From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics.” > — Richard Feynman · *The Feynman Lectures on Physics*, Volume II, chapter 1, 1964 [r-11.19](ch11-refs.md#r-11.19) Formula: ([∫₀¹2x dx − 1](ch11-cards.md#g-ftc))/([(x²)′(1)](ch11-cards.md#g-dsq1)) {brace: [calculus](ch11.md#ch-T11-calculus)} · ([(∂_t−∂_xx)[e^−π²tsin(πx)]](ch11-cards.md#g-heatzero))/([π²](ch11-cards.md#g-decayrate)) {brace: [differential equations](ch11.md#ch-T11-differential-equations)} · ([min_x∈ℝx²](ch11-cards.md#g-minimumzero))/([min_x∈ℝ(x² + 1)](ch11-cards.md#g-minplus1)) {brace: [optimization](ch11.md#ch-T11-optimization)} · ([∇·(∇×𝐯)](ch11-cards.md#g-divcurl) + [∇×(∇f)](ch11-cards.md#g-curlgrad) + [∇²(1/r), r ≠ 0](ch11-cards.md#g-laplace1r) + [d(dω)](ch11-cards.md#g-ddzero))/(([∇·𝐫](ch11-cards.md#g-divr) + [∇²(r²)](ch11-cards.md#g-laprsq)) {brace: [divergence](ch11-cards.md#g-nabla)} · ([∮_S² (𝐫̂/r²)·d𝐀](ch11-cards.md#g-flux) + [½∮_S¹(x dy − y dx)](ch11-cards.md#g-greenarea)) {brace: [flux and circulation](ch11-cards.md#g-stokes)} ) {brace: [vector calculus](ch11.md#ch-T11-vector-calculus)} #### Type of number Proven transcendental · Key concepts: Calculus, differential equations, optimization, and vector calculus. It includes symbols you may not have seen before, such as: [∫](ch11-cards.md#g-ftc), [∇](ch11-cards.md#g-nabla), and [∮](ch11-cards.md#g-flux). Zero here is balance: change undoing accumulation, a cooling rod obeying its equation exactly, the floor of a bowl, and a whirl with nothing flowing out. We have five rooms to explore, and I’m starting us by the water. The fountain feeds a flow tank; watch the water pass a fixed point. A metal rod cools behind glass nearby, and a bowl-shaped model offers slopes to inspect. This gallery begins with motion; shape, chance, symmetry, and overlap are waiting in the others. Field notes ### [calculus](ch11.md#ch-T11-calculus) {#ch-T11-calculus} The local question, “what is the slope here?”, and the accumulated one, “how much has the function changed between these endpoints?”, joined by one theorem. #### Numerator [∫₀¹2x dx − 1](ch11-cards.md#g-ftc) = 0; the [fundamental theorem of calculus](ch11-cards.md#g-ftc), the result the whole subject rests on, applied to x². The derivative of x² is 2x, and integrating it from 0 to 1 recovers the change in x²: 1² − 0² = 1. Subtracting the 1 in the numerator leaves nothing. Newton and Leibniz helped make that connection into a systematic method. [r-11.1](ch11-refs.md#r-11.1) [r-11.23](ch11-refs.md#r-11.23) #### Denominator [(x²)′(1)](ch11-cards.md#g-dsq1) = 2; the slope at the right endpoint of the same parabola. The prime means derivative and the (1) says where to evaluate it: 2x at x = 1 is 2. A slope can certainly be zero elsewhere; we chose the point where it is two. [r-11.2](ch11-refs.md#r-11.2) The first fraction is 0/2. ### [differential equations](ch11.md#ch-T11-differential-equations) {#ch-T11-differential-equations} A differential equation specifies a relationship among a function and its rates of change; solving it means finding a function that makes the relationship hold. The cooling rod gives us a change we can follow over time. #### Numerator [(∂_t−∂_xx)[e^−π²tsin(πx)]](ch11-cards.md#g-heatzero) = 0; the heat equation on a rod simple enough to solve all the way through. Take a rod of length 1, held at zero temperature at both ends, in units that set its diffusivity, the constant that fixes how fast heat spreads through it, to 1. Write u for the temperature at position x and time t. The heat equation then reads uₜ = uₓₓ: how fast the temperature changes in time (uₜ, the ∂_t of the formula) equals how sharply the temperature curve bends along the rod (uₓₓ, the ∂_xx). One solution is [u(x, t) = e^−π²t sin(πx)](ch11-cards.md#g-heatzero): the sine vanishes at x = 0 and x = 1, and the exponential lowers the temperature while keeping that shape. Differentiating once in time gives −π²u; twice in space also gives −π²u. The two cancel, so (∂_t−∂_xx)u, the change in time minus the bending, is zero throughout the rod. That zero belongs to the equation, not to the rod’s temperature; the rod is still cooling, and the zero says it is cooling exactly as the equation requires. [r-11.3](ch11-refs.md#r-11.3) #### Denominator [π²](ch11-cards.md#g-decayrate) ≈ 9.8696; the rate at which that same temperature pattern fades. In these normalized units the solution decays like e^−π²t, so [π²](ch11-cards.md#g-decayrate) is its decay rate, positive and (because π is) transcendental. [r-11.4](ch11-refs.md#r-11.4) The second fraction is 0/[π²](ch11-cards.md#g-decayrate). **Fine print.** A different rod length or diffusivity would change the physical decay rate. ### [optimization](ch11.md#ch-T11-optimization) {#ch-T11-optimization} Finding the lowest point: a bowl touching the floor, a simplified valley whose bottom we can locate exactly. #### Numerator [min_x∈ℝx²](ch11-cards.md#g-minimumzero) = 0; the simplest problem optimization has. x² is never negative and reaches zero at x = 0, so its least value is 0. Calculus finds the candidate where the derivative 2x is zero, but [a zero derivative](ch11-cards.md#g-fermatmax) alone could also mark a maximum or neither; we know it is the global minimum because x² ≥ 0 everywhere. [r-11.5](ch11-refs.md#r-11.5) #### Denominator [min_x∈ℝ(x² + 1)](ch11-cards.md#g-minplus1) = 1; the same bowl, lifted by one. x² + 1 has its minimum at the same place, x = 0, but the least value is now 1. Moving a function upward changes its least value without changing where that value occurs. The third fraction is 0/1. **Fine print.** The minimum value and the minimizing location are different answers; for x² both happen to be 0. The denominator uses the value, not the location, so it is safely nonzero. ### [vector calculus](ch11.md#ch-T11-vector-calculus) {#ch-T11-vector-calculus} Fields put a value or an arrow at every point: a temperature field assigns a number, a velocity field an arrow. They are differentiated with [∇](ch11-cards.md#g-nabla), nabla, which bundles the derivatives in the three directions (∂/∂x, ∂/∂y, ∂/∂z), and the way it is attached says what it measures: dotted, ∇·, it gives the divergence, a field’s local net outflow; crossed, ∇×, it gives the curl, the field’s local rotation. Electromagnetism made fields indispensable: [Maxwell’s equations](ch11-cards.md#g-maxwell) are four statements about the divergence and curl of two fields. [r-11.13](ch11-refs.md#r-11.13) #### Numerator [∇·(∇×𝐯)](ch11-cards.md#g-divcurl) = 0; a fundamental identity: a curl has no net outflow. Take a field whose second derivatives are continuous and find its curl, the field of little whirls inside it; then ask for the divergence of that curl, its net outflow. Written in coordinates, every term turns up twice with opposite signs, once as ∂_x∂_y and once as ∂_y∂_x; because a [mixed second derivative](ch11-cards.md#g-clairaut) does not depend on the order of differentiation, each pair cancels exactly. A whirl in the flow tank is a picture of curl, and the identity is about the [divergence of a curl](ch11-cards.md#g-divcurl): the whirls themselves have no net outflow, which is not the same as saying the swirling water has none. [r-11.6](ch11-refs.md#r-11.6) [∇×(∇f)](ch11-cards.md#g-curlgrad) = 0; the twin identity: a landscape of heights has no whirlpools. The gradient ∇f points in the direction of steepest increase, like arrows pointing uphill on a map of heights. If f has continuous second derivatives, its gradient has no curl: walk any closed loop on the hillside and you finish at the height you started from, so the uphill arrows can never add up to a circulation round the loop. [r-11.6](ch11-refs.md#r-11.6) [∇²(1/r), r ≠ 0](ch11-cards.md#g-laplace1r) = 0; the potential of a point charge or mass, [harmonic](ch11-cards.md#g-laplaceeq) in empty space. The Laplacian ∇² adds up the second derivatives in the three coordinate directions, and a function whose Laplacian is zero is called harmonic. [r-11.6](ch11-refs.md#r-11.6) For 1/r in three-dimensional space the three sum to zero everywhere except at the source itself, where r = 0 and 1/r has no value; that is why r ≠ 0 is written into the term. [r-11.7](ch11-refs.md#r-11.7) [d(dω)](ch11-cards.md#g-ddzero) = 0; d² = 0, the single identity behind the first two. Differential forms are what one integrates over curves, surfaces, and higher-dimensional regions, and the exterior derivative d is the way to differentiate them. Applying it twice always gives zero, and in three dimensions that one fact unpacks into [div curl = 0](ch11-cards.md#g-divcurl) and [curl grad = 0](ch11-cards.md#g-curlgrad). In the next chapter we meet its partner: [a boundary has no further boundary](ch12-cards.md#g-bdbd). [r-11.16](ch11-refs.md#r-11.16) #### Denominator Two groups, multiplied: local spreading and bending, then flow across a surface and area enclosed by a walk. In three dimensions, r = (x, y, z) and r² = x² + y² + z². divergence [∇·𝐫](ch11-cards.md#g-divr) = 3; the divergence of position itself. Each coordinate, differentiated by itself, gives 1: ∂x/∂x + ∂y/∂y + ∂z/∂z = 1 + 1 + 1 = 3. [r-11.8](ch11-refs.md#r-11.8) [∇²(r²)](ch11-cards.md#g-laprsq) = 6; the Laplacian of squared distance. Each of x², y², z² contributes a second derivative of 2. flux and circulation [∮_S² (𝐫̂/r²)·d𝐀](ch11-cards.md#g-flux) = 4π; what crosses a surface: the solid angle of the whole sky. The loop on ∮ means the integral is taken all the way round something closed: here the sphere S², and in the next term the unit circle. [r-11.31](ch11-refs.md#r-11.31) Differentiating the 1/r potential above gives, up to sign, exactly this inverse-square field r̂/r², and it weakens at exactly the rate a centered sphere’s area grows, so [the outward flux is 4π](ch11-cards.md#g-flux) whatever the radius. [r-11.9](ch11-refs.md#r-11.9) The [divergence theorem](ch11-cards.md#g-divthm) keeps the books: what flows out through a closed surface equals the total source inside. [r-11.10](ch11-refs.md#r-11.10) Away from the center the field’s divergence is zero, since up to sign it is the ∇²(1/r) above; all of the 4π comes from the point source at the center, which every such sphere encloses. [r-11.7](ch11-refs.md#r-11.7) [½∮_S¹(x dy − y dx)](ch11-cards.md#g-greenarea) = π; what a closed walk encloses, by [Green’s theorem](ch16-cards.md#g-greenthm). Walking once counterclockwise round the unit circle and integrating measures the area inside: the disk’s area, π. [r-11.11](ch11-refs.md#r-11.11) The fourth and final fraction is 0/(45π): (3 + 6) × (4π + π) = 9 × 5π = 45π. **Fine print.** The numerator mixes scalar fields, a vector field, and a differential form; each is evaluated first and its zero enters as the scalar 0. Orientation matters for [Green’s theorem](ch16-cards.md#g-greenthm): clockwise would reverse the sign. What this chapter adds to the formula: 0/2 · 0/[π²](ch11-cards.md#g-decayrate) · 0/1 · 0/(45π) = 0 We have measured slopes, followed heat, found a minimum, and asked how much flows out through a closed surface. Come through this doorway with me. There’s rope on the next table, along with some surfaces that reward a closer look. We have been watching what flows through a surface; now I want to show you what happens when we pick the surface itself up and try to change its shape. Watch and read - [Divergence and curl: The language of Maxwell’s equations, fluid flow, and more](https://www.youtube.com/watch?v=rB83DpBJQsE) 3Blue1Brown; the two operators drawn as sources and whirlpools, which is the only way to see why one kills the other - [Maxwell’s Equations Explained Intuitively](https://www.youtube.com/watch?v=rvb6KVwasd8) all four equations in words and pictures, including ∇·B = 0, the law this chapter’s zero is the shadow of - [Vector calculus identities](https://en.wikipedia.org/wiki/Vector_calculus_identities) Wikipedia - [Divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem) Wikipedia