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# Glossary cards — Chapter ᚉ(C): The gallery of motion

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T11, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch11.md)

*Calculus, differential equations, optimization, and vector calculus: the fundamental theorem, a cooling rod, the bottom of a bowl, divergence and curl. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Coll — `ᚉ` {#g-og-coll}

*Ogham letter*

Ogham letter name meaning ‘hazel’, the fairest and the sweetest of the trees in its kennings.

Coll means ‘hazel-tree’, cognate with Welsh collen and with Latin corylus, and is the fifth of the transparent tree names. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give caíniu fedaib, “fairest tree”; carae blóesc, “friend of nutshells”; and milsem fedo, “sweetest tree.” The Auraicept’s epithet for it, cainfidh, fair-wood, is apt, and it adds the plainest of its remarks: “coll, that is, every one is eating from its nuts.” Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19)

### the fundamental theorem of calculus, for x² — `∫_{0}^{1}2x dx − 1` {#g-ftc}

*Integer or rational*

0: the integral of the derivative of x² from 0 to 1 is x² evaluated at the ends, 1 − 0, so the difference vanishes.

The fundamental theorem joins the two operations that created calculus. If f is continuously differentiable on [a, b] then ∫_{a}^{b}f′(x)dx = f(b) − f(a); for f(x) = x² on [0, 1] the integral of 2x is 1 and f(1) − f(0) is 1, so the residual is 0. Newton and Leibniz developed the reciprocal relationship independently in the late seventeenth century. The instance on the page needs only ordinary one-variable calculus.

Cites: [11.1](ch11-refs.md#r-11.1), [11.23](ch11-refs.md#r-11.23), [11.24](ch11-refs.md#r-11.24)

### the derivative of x² at one — `(x²)′(1)` {#g-dsq1}

*Integer or rational*

2: the slope of the parabola at x = 1, and the number the numerator’s integral is built from.

The derivative of x² is 2x, so at x = 1 it is 2. Newton called it a fluxion and Leibniz wrote dy/dx; Lagrange’s prime notation, f′, dates from 1797. An integer.

Cites: [11.1](ch11-refs.md#r-11.1), [11.2](ch11-refs.md#r-11.2)

### an exact heat-equation solution — `(∂_{t}−∂_{xx})[e^{−π²t}sin(πx)]` {#g-heatzero}

*Integer or rational*

Exactly 0: time decay and spatial curvature cancel for this solution of uₜ = uₓₓ.

Let u(x, t)=e^{−π²t}sin(πx). Then ∂_{t}u=−π²u and ∂_{xx}u=−π²u, so (∂_{t}−∂_{xx})u=0. Fourier used such separated modes in his 1807–1822 theory of heat.

Cites: [11.3](ch11-refs.md#r-11.3)

### the decay rate of the first heat mode — `π²` {#g-decayrate}

*Proven transcendental*

About 9.8696: the mode sin(πx) of the heat equation on a unit rod decays like e^(−π²t), and π² is transcendental because π is.

Fourier’s method, from his 1807 memoir and his Théorie analytique de la chaleur of 1822, separates the heat equation u_{t} = u_{xx} into modes sin(nπx) that each decay at their own rate n²π²; the slowest, n = 1, sets how long a rod takes to cool. π² ≈ 9.8696 is transcendental by Lindemann’s 1882 theorem, since if π² were algebraic, its square root π would be too. It is also ζ(2)·6, which is how Chapter ᚑ(O) knows it.

Cites: [11.3](ch11-refs.md#r-11.3), [11.4](ch11-refs.md#r-11.4)

See also: [the Basel value](ch18-cards.md#g-zeta2)

### the simplest convex minimum — `min_{x∈ℝ}x²` {#g-minimumzero}

*Integer or rational*

Exactly 0, attained uniquely at x = 0: the simplest problem optimization has, and the bottom of the bowl.

The real function x² is nonnegative and equals 0 only at x=0, so min_{x∈ℝ}x²=0. The example is elementary, but its notation opens the framework of mathematical optimization: objective functions, feasible sets, convexity, gradients, duality, and optimality conditions.

Cites: [11.5](ch11-refs.md#r-11.5)

### the minimum of x squared plus one — `min_{x∈ℝ}(x² + 1)` {#g-minplus1}

*Integer or rational*

1, at x = 0: the same bowl as the numerator, lifted by one so that the least value is no longer nothing.

The function x² + 1 is convex, its derivative 2x vanishes only at x = 0, and its value there is 1. Fermat’s method for finding maxima and minima, setting the rate of change to zero, circulated in about 1636 and calculus-based rules for finding optima begin with it and with Lagrange. An integer.

Cites: [11.17](ch11-refs.md#r-11.17), [11.5](ch11-refs.md#r-11.5)

See also: [Fermat’s method of maxima and minima](ch11-cards.md#g-fermatmax)

### Fermat’s method of maxima and minima — `f′(x) = 0` {#g-fermatmax}

*Function or operator*

Locate an extremum by setting the rate of change to zero: Fermat’s method, from a Latin treatise of about 1636, before there was a differential calculus to put it in.

Pierre de Fermat’s Methodus ad disquirendam maximam et minimam circulated in about 1636. The procedure it describes, adequality, compares the value of a quantity at a point with its value at a neighboring point and then lets the difference vanish, which is in substance the rule that at a maximum or a minimum the rate of change is zero. It predates the differential calculus that would later justify it, and the calculus-based rules for finding optima begin with it and with Lagrange. A method, not a number.

Cites: [11.17](ch11-refs.md#r-11.17)

### divergence of a curl — `∇·(∇×𝐯)` {#g-divcurl}

*Integer or rational*

Identically 0 for any smooth field, because mixed partial derivatives commute.

The identity ∇·(∇×v) = 0 holds for every field whose second partial derivatives are continuous, by Clairaut’s theorem that mixed partials commute, proposed in 1740 and proved by Schwarz in 1873; it became a one-line identity once Heaviside and Gibbs put ∇ to work as grad, div, and curl in the 1880s, when Maxwell’s 1865 equations were rewritten in that notation. On all of ℝ³, or on a suitable region without holes, it is the compatibility condition for writing a divergence-free field such as the magnetic field as the curl of a vector potential; in the language of differential forms it is half of d² = 0. Its value is the number 0 at every point, an integer.

Cites: [11.13](ch11-refs.md#r-11.13), [11.14](ch11-refs.md#r-11.14), [11.18](ch11-refs.md#r-11.18), [11.6](ch11-refs.md#r-11.6)

See also: [Clairaut’s theorem](ch11-cards.md#g-clairaut), [the exterior derivative applied twice](ch11-cards.md#g-ddzero), [Maxwell’s equations](ch11-cards.md#g-maxwell), [the vector potential](ch11-cards.md#g-vectorpot)

### divergence of the position field — `∇·𝐫` {#g-divr}

*Integer or rational*

3, one for each dimension.

The position field r = (x, y, z) has divergence ∂x/∂x + ∂y/∂y + ∂z/∂z = 3, one for each dimension; in n dimensions it is n. It is the simplest example of a field with a source everywhere, and it appears in the divergence theorem as the trick that turns a volume into a surface integral: the volume of a region is one third of the flux of r through its boundary. An integer, 3.

Cites: [11.8](ch11-refs.md#r-11.8)

See also: [the divergence theorem](ch11-cards.md#g-divthm), [one third](ch04-cards.md#g-third)

### Gauss’s flux integral — `∮_{S²} (𝐫̂/r²)·d𝐀` {#g-flux}

*Proven transcendental*

4π, the solid angle of the whole sky.

Gauss’s law is the first of the lemmas Carl Friedrich Gauss proved in his Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata, published in 1813 and written on the attraction of ellipsoids. Lagrange’s memoir of 1773 supplies the change of variables the proof runs on, but it contains no flux theorem: the memoir has been read end to end, and there is no closed surface in it, no flux, and no 4π. Gauss came back to inverse-square forces in general in his Allgemeine Lehrsätze of 1840. The law says the flux of an inverse-square field through any closed surface around its source is 4π times the source strength; for the unit field r̂/r² through a sphere it is exactly 4π, the solid angle of all directions. It is one of Maxwell’s equations, and it is essentially equivalent to the inverse-square laws of Coulomb and Newton. Transcendental, since π is, by Lindemann’s theorem of 1882.

Cites: [11.22](ch11-refs.md#r-11.22), [11.25](ch11-refs.md#r-11.25), [11.26](ch11-refs.md#r-11.26), [11.9](ch11-refs.md#r-11.9)

See also: [Maxwell’s equations](ch11-cards.md#g-maxwell)

### curl of a gradient — `∇×(∇f)` {#g-curlgrad}

*Integer or rational*

Identically 0 when f has continuous second derivatives: a landscape of heights has no whirlpools, the twin of the divergence of a curl.

∇×(∇f) = 0 for any twice continuously differentiable f, again because mixed partials commute, which Clairaut proposed in 1740 and Schwarz proved in 1873. So a conservative force field, one that is a gradient of a potential, has no curl, and it does no net work around any loop; the identity is the other half of d² = 0 in the language of forms. The zero vector at every point, handed home as the integer 0.

Cites: [11.6](ch11-refs.md#r-11.6)

See also: [the exterior derivative applied twice](ch11-cards.md#g-ddzero), [divergence of a curl](ch11-cards.md#g-divcurl)

### the 1/r potential is harmonic — `∇²(1/r), r ≠ 0` {#g-laplace1r}

*Integer or rational*

The Laplacian of 1/r is 0 everywhere except at the origin: the potential of a point charge or mass satisfies Laplace’s equation in empty space.

Pierre-Simon Laplace put the equation at the center of potential theory in his memoir on the attraction of spheroids, in the Mémoires volume for 1782, writing it in polar rather than Cartesian coordinates; it was known before him, and it carries his name because he made it the working tool of the subject. 1/r solves it away from the origin, the distributional identity at the origin is ∇²(1/r) = −4πδ, and Siméon Denis Poisson showed in ~1813 that inside the attracting matter the potential satisfies ∇²V = −4πρ instead, which is Gauss’s law again. The Laplacian ∇² = ∇·∇ is the sum of second derivatives. Exactly 0 for r ≠ 0, an integer.

Cites: [11.21](ch11-refs.md#r-11.21), [11.7](ch11-refs.md#r-11.7)

See also: [Laplace’s equation](ch11-cards.md#g-laplaceeq)

### the exterior derivative applied twice — `d(dω)` {#g-ddzero}

*Integer or rational*

The exterior derivative applied twice is 0, d² = 0: the one identity from which div curl = 0 and curl grad = 0 both follow.

Élie Cartan formalized the exterior derivative in 1899; the identity d∘d = 0 holds because mixed partials commute, and in three dimensions it unpacks to curl grad = 0 and div curl = 0. Its dual statement is that the boundary of a boundary is empty, ∂∂ = 0, the axiom of homology in Chapter ᚊ(Q); Stokes’ theorem ∫_{M} dω = ∫_{∂M} ω links the two. The value is the zero form, handed home as the integer 0.

Cites: [11.16](ch11-refs.md#r-11.16)

See also: [the boundary of a boundary](ch12-cards.md#g-bdbd), [curl of a gradient](ch11-cards.md#g-curlgrad), [divergence of a curl](ch11-cards.md#g-divcurl), [homology groups](ch12-cards.md#g-homology), [Stokes’ theorem](ch11-cards.md#g-stokes)

### the Laplacian of r squared — `∇²(r²)` {#g-laprsq}

*Integer or rational*

6 in three dimensions: each of x², y², z² contributes a second derivative of 2.

For r² = x² + y² + z² the Laplacian is 2 + 2 + 2 = 6; in n dimensions it is 2n. It is the reason the mean value of a function over a small sphere exceeds its center value by ∇²f·r²/6, to leading order, in three dimensions. An integer.

Cites: [11.8](ch11-refs.md#r-11.8)

### the area of the unit circle by Green’s theorem — `½∮_{S¹}(x dy − y dx)` {#g-greenarea}

*Proven transcendental*

π: walking once round the unit circle and integrating measures the area inside, Green’s theorem.

George Green, a self-taught miller’s son from Nottingham, published a version of his theorem in 1828 in an essay he paid to print himself and sold to fifty-one subscribers, most of them members of the Nottingham Subscription Library; it went almost unnoticed until William Thomson, later Lord Kelvin, was given three copies in 1845, showed it to Liouville and Sturm in Paris that summer, and had it reprinted between 1850 and 1854. The theorem turns an integral round a boundary into an integral over the region inside, and the special case ½∮(x dy − y dx) gives the enclosed area, which for the unit circle is π. It is the two-dimensional case of Stokes’ theorem. Transcendental, because π is.

Cites: [11.11](ch11-refs.md#r-11.11), [11.20](ch11-refs.md#r-11.20), [11.9](ch11-refs.md#r-11.9)

See also: [pi](ch04-cards.md#g-pi), [Stokes’ theorem](ch11-cards.md#g-stokes)

### nabla — `∇` {#g-nabla}

*Function or operator*

The vector of partial derivatives (∂/∂x, ∂/∂y, ∂/∂z): dotted for divergence, crossed for curl, applied for gradient, squared for the Laplacian.

William Rowan Hamilton introduced the operator, writing it as a sideways ◁, and Peter Guthrie Tait fixed the upright ∇ as the standard symbol in his Elementary Treatise on Quaternions of 1867; the name nabla, after a Phoenician harp of that shape, by way of the Greek νάβλα, was proposed to Tait by William Robertson Smith in a letter of 10 November 1870. Josiah Willard Gibbs and Oliver Heaviside separated it from the quaternions in the 1880s into the modern grad, div, and curl. An operator, not a number.

Cites: [11.15](ch11-refs.md#r-11.15), [11.26](ch11-refs.md#r-11.26), [11.6](ch11-refs.md#r-11.6), [22.15](ch22-refs.md#r-22.15)

See also: [the quaternions](ch22-cards.md#g-quaternions)

### the divergence theorem — `∫_{V} ∇·𝐅 dV = ∮_{∂V} 𝐅·d𝐀` {#g-divthm}

*Function or operator*

What flows out through a surface equals the total source inside; special cases Gauss ~1813, the general theorem Ostrogradsky 1826, and the attribution is contested.

Carl Friedrich Gauss proved three special cases in 1813, in a paper on the attraction of ellipsoids, and Mikhail Ostrogradsky gave the general theorem in 1826, published in 1831. It is the three-dimensional case of the general Stokes’ theorem and the reason Gauss’s flux integral in the denominator stays 4π through any closed surface around the source. A theorem, not a number.

Cites: [11.10](ch11-refs.md#r-11.10)

See also: [who the divergence theorem belongs to](ch11-cards.md#g-divattrib), [Gauss’s flux integral](ch11-cards.md#g-flux), [Stokes’ theorem](ch11-cards.md#g-stokes)

### Stokes’ theorem — `∮_{∂S} 𝐅·d𝐫 = ∫_{S} (∇×𝐅)·d𝐀` {#g-stokes}

*Function or operator*

Circulation round a loop equals the curl through any surface it bounds; Kelvin sent it to Stokes in 1850 and Stokes set it as an exam question in 1854.

William Thomson (Lord Kelvin) wrote the theorem in a letter of 1850 and George Stokes put it on the Smith’s Prize examination at Cambridge in 1854, which is how it got his name; Maxwell sat that exam. The general form ∫_{M} dω = ∫_{∂M} ω is Élie Cartan’s, after earlier generalizations by Vito Volterra, Édouard Goursat, and Henri Poincaré; it contains Green’s theorem, the divergence theorem, and the fundamental theorem of calculus as special cases. A theorem, not a number.

Cites: [11.12](ch11-refs.md#r-11.12)

See also: [the divergence theorem](ch11-cards.md#g-divthm)

### Maxwell’s equations — `∇·𝐄 = ρ/ε₀, ∇·𝐁 = 0, ∇×𝐄 = −∂𝐁/∂t, ∇×𝐁 = μ₀𝐉 + μ₀ε₀∂𝐄/∂t` {#g-maxwell}

*Function or operator*

The four equations of electromagnetism, 1865, in the vector form Heaviside gave them in 1884: the theory in which div and curl were first studied.

James Clerk Maxwell published twenty equations in 1865; Oliver Heaviside rewrote them as these four in 1884 using the new vector calculus. They predict light as an electromagnetic wave with speed 1/√(μ₀ε₀), as Maxwell had found in 1865. The second equation, ∇·𝐁 = 0, is the zero of Chapter ᚁ(B)’s laws brace, and it is consistent with this chapter’s numerator because 𝐁 is the curl of a vector potential. Four equations, not a number.

Cites: [11.13](ch11-refs.md#r-11.13)

See also: [no magnetic monopoles](ch03-cards.md#g-divB), [the vector potential](ch11-cards.md#g-vectorpot)

### Clairaut’s theorem — `∂²f/∂x∂y = ∂²f/∂y∂x` {#g-clairaut}

*Function or operator*

Mixed partial derivatives commute wherever the second partials are continuous; Alexis Clairaut proposed it in 1740 and H. A. Schwarz gave the first rigorous proof in 1873.

The brace’s two curl identities are this one theorem in disguise. Writing out ∇·(∇×v) or ∇×(∇f) gives terms that appear twice with opposite signs, once as ∂_{x}∂_{y} and once as ∂_{y}∂_{x}, so they cancel exactly rather than approximately. The hypothesis is not decoration: the equality is asserted where the second partial derivatives are continuous on a neighborhood of the point, which is why the chapter states the twin identity for functions whose second derivatives are continuous. Clairaut proposed the result in 1740; the first rigorous proof is Schwarz’s, in 1873. A theorem, not a number.

Cites: [11.6](ch11-refs.md#r-11.6)

See also: [curl of a gradient](ch11-cards.md#g-curlgrad)

### Laplace’s equation — `∇²V = 0` {#g-laplaceeq}

*Function or operator*

The equation a potential satisfies in empty space; inside the matter itself Poisson’s correction of ~1813 replaces it with ∇²V = −4πρ.

The function 1/r solves ∇²V = 0 everywhere except at the origin, which is why the potential of a point charge or a point mass is harmonic in empty space; at the origin itself the distributional identity is ∇²(1/r) = −4πδ. Pierre-Simon Laplace made the equation the working tool of potential theory in his memoir on the attraction of spheroids, writing it in polar rather than Cartesian coordinates; the equation was known before him, and it carries his name because he made it the instrument of the subject. Siméon Denis Poisson showed in ~1813 that inside the attracting matter the potential satisfies ∇²V = −4πρ instead, so that the whole of the missing divergence sits where the mass or charge is. An equation, not a number.

Cites: [11.21](ch11-refs.md#r-11.21), [11.7](ch11-refs.md#r-11.7)

See also: [the 1/r potential is harmonic](ch11-cards.md#g-laplace1r)

### the vector potential — `B = ∇×A` {#g-vectorpot}

*Function or operator*

A field whose curl is the field you started with: a smooth enough divergence-free field on the whole of space has one, but on a region with a point taken out it may not.

Because the divergence of a curl is 0, a field can be a curl only if its divergence vanishes. On all of ℝ³ that condition is also sufficient for a field that is twice continuously differentiable and decays at least as fast as 1/‖x‖. On a smaller region it depends on the shape of the region, and simple connectivity is not enough: ℝ³ with the origin removed is simply connected, yet the inverse-square field r̂/r² has zero divergence on it and no vector potential. What suffices is that the region be star-shaped or contractible, by the Poincaré lemma, and in general that its second de Rham cohomology vanish. The magnetic field is the physical instance, and in quantum mechanics A acts on a charged particle even where the magnetic field is zero, the Aharonov–Bohm effect. A field, not a number.

Cites: [11.14](ch11-refs.md#r-11.14), [11.18](ch11-refs.md#r-11.18)

See also: [cohomology](ch12-cards.md#g-cohomology), [divergence of a curl](ch11-cards.md#g-divcurl)

### who the divergence theorem belongs to — `1762 … 1839` {#g-divattrib}

*Function or operator*

One theorem, six names and four dates: Gauss ~1813, Ostrogradsky 1826, Green 1828, and others before them, and this book takes the earliest it can source.

Six names are attached to this one identity and the reference works do not agree on which belongs on it. The Encyclopedia of Mathematics lists it under all six at once—Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula, and Gauss-Green-Ostrogradski formula—and adds that it is already present in the work of Euler and other eighteenth-century mathematicians. The dates in circulation are these. Lagrange introduced surface integrals in 1760 and is widely credited with the divergence theorem itself in 1762, and that credit does not survive being checked: his 1762 memoir integrates by parts and then disposes of the boundary terms, and Victor Katz’s history names him once, among those who used the fundamental theorem to reduce an integral by a dimension, and not as an author of the result. Katz’s bibliography does not list him at all. Gauss proved three special cases in 1813, while working on the gravitational attraction of an elliptical spheroid, and further special cases in 1833 and 1839. Mikhail Ostrogradsky submitted Demonstration d’un theoreme du calcul integral to the Paris Academy on 13 February 1826, stating and proving the general theorem; he presented it again in Paris on 6 August 1827 and in St Petersburg on 5 November 1828, and only the last of the three was published, in 1831, in Note sur la Theorie de la Chaleur. George Green proved special cases in 1828 in his Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Gauss’s law, which is the inverse-square statement rather than the general theorem, used to carry a pair of dates here too: Lagrange in 1773 and Gauss in 1835. Both were checked in September 2026 and both went. Lagrange’s 1773 memoir was read end to end and contains no flux theorem, no closed surface, and no 4π; what it supplies is the change of variables Gauss’s proof uses. Nothing at all could be found at 1835, and the citation hanging off that year in the reference work that prints it is the 1813 paper. Gauss’s law is Gauss’s, in 1813. That still leaves four dates for the theorem, in a chapter that gives Clairaut a year and Green a year and Poisson a year without anyone objecting. The disagreement is nobody’s mistake. A result that is stated for one special case, proved for another, generalized by a third man, presented three times, and published five years after its first presentation has no single date to be right about, and the names it collected record which paper each later school of physics happened to read first. This book does not adjudicate priority. Where the prose needs a date it uses the earliest it can source—Gauss ~1813 for the divergence theorem, Gauss 1813 for Gauss’s law—and the rest of the story lives here.

Cites: [11.10](ch11-refs.md#r-11.10), [11.22](ch11-refs.md#r-11.22), [11.25](ch11-refs.md#r-11.25)

See also: [the divergence theorem](ch11-cards.md#g-divthm)