# References — Chapter ᚉ(C): The gallery of motion *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-11, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch11.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 11.1 {#r-11.1} The fundamental theorem gives ∫_a^b f′(x) dx = f(b) − f(a) under the usual continuity hypotheses; for x² on [0, 1], the integral of 2x and the endpoint difference are both 1. *Status: confirmed* - [Wolfram MathWorld, Fundamental Theorems of Calculus](https://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html) — states the endpoint formula ∫_a^b f(x)dx = F(b) − F(a) for a continuous integrand and antiderivative. ### 11.2 {#r-11.2} The prime derivative notation is associated with Lagrange and appears in his 1797 Théorie des fonctions analytiques; Newton used fluxions and Leibniz differential notation. *Status: confirmed* - [MacTutor, Earliest Uses of Symbols of Calculus](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/) — records Lagrange’s f′ notation in 1797 and distinguishes earlier Newton/Leibniz notations. - [Fluxion, Wikipedia](https://en.wikipedia.org/wiki/Fluxion) — “Fluxions were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to time).” ### 11.3 {#r-11.3} For u=e^(−π²t)sin(πx), the heat operator (∂t−∂xx)u is zero; Fourier’s heat theory dates to his 1807 work and the separated modes on a unit rod decay at rates n²π². Two events: the 1807 memoir already solved the heat equation with trigonometric series (MacTutor), and the Heat equation page dates the separation-of-variables solution to the Théorie analytique de la chaleur of 1822. *Status: confirmed* - [MacTutor, Joseph Fourier](https://mathshistory.st-andrews.ac.uk/Biographies/Fourier/) — Fourier completed his memoir On the Propagation of Heat in Solid Bodies by 1807 and read it to the Paris Institute on 21 December 1807; “He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions,” and Lagrange and Laplace’s first objection, in 1808, “was to Fourier’s expansions of functions as trigonometrical series.” (The review first cited here: T. N. Narasimhan, Fourier’s heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1999) 151–172, doi:10.1029/1998RG900006, paywalled). - [Wikipedia, Heat equation](https://en.wikipedia.org/wiki/Heat_equation) — gives Fourier’s heat equation and his separation-of-variables solution, proposed in his Théorie analytique de la chaleur (1822). - **Own computation** (the author's own work) — Differentiate directly: ∂t u = −π²u and ∂xx u = −π²u, so their difference is 0; for sin(nπx), the same calculation gives decay rate n²π². ### 11.4 {#r-11.4} π² is approximately 9.8696, is transcendental, and equals 6ζ(2). *Status: confirmed* - [Wolfram MathWorld, Pi](https://mathworld.wolfram.com/Pi.html) — states that π is transcendental by Lindemann’s 1882 result. - [NIST Digital Library of Mathematical Functions §25.6](https://dlmf.nist.gov/25.6) — equation 25.6.1 gives ζ(2)=π²/6, hence π²=6ζ(2). ### 11.5 {#r-11.5} The real minimum of x² is 0, attained uniquely at x=0, and the minimum of x²+1 is 1 at x=0. The framework of mathematical optimization names these parts: the objective function, the search space or choice set, convexity and duality. *Status: confirmed* - **Own computation** (the author's own work) — For every real x, x²≥0 with equality exactly at x=0; adding 1 shifts every value by 1, so the unique minimizer remains x=0 and the minimum becomes 1. - [Mathematical optimization, Wikipedia](https://en.wikipedia.org/wiki/Mathematical_optimization) — its notation example: “The minimum value in this case is 1, occurring at x = 0.”; and the vocabulary: “The function f is variously called an objective function,” “The domain A of f is called the search space or the choice set,” “When the objective function is a convex function, then any local minimum will also be a global minimum,” and “the theory of duality.” ### 11.6 {#r-11.6} For fields whose second partial derivatives are continuous, div(curl v)=0 and curl(grad f)=0, because mixed partial derivatives commute. That mixed partials commute is Clairaut’s theorem, proposed by Alexis Clairaut in 1740 and first proved rigorously by H. A. Schwarz in 1873. Both identities are the vector-calculus form of d²=0. Vector calculus identities proves curl grad = 0 by the equality of mixed partials and calls div curl = 0 a special case of d² = 0. A conservative field, the gradient of a function, is irrotational and does zero work around any simple closed loop. The gradient points in the direction of most rapid increase, and a function whose Laplacian is 0 is called harmonic. *Status: confirmed* - [Wolfram MathWorld, Vector Derivative](https://mathworld.wolfram.com/VectorDerivative.html) — lists the standard gradient, divergence, curl, and Laplacian operators and their Cartesian derivative identities. - [Wolfram MathWorld, Stokes’ Theorem](https://mathworld.wolfram.com/StokesTheorem.html) — identifies the gradient, curl, and divergence theorems as differential-form instances of Stokes’ theorem. - [Wikipedia, Symmetry of second derivatives](https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives) — records Clairaut’s proposed proof of 1740 and Schwarz’s first rigorous proof of 1873, and states the hypothesis as continuous second partial derivatives on a neighborhood of the point. - [Vector calculus identities, Wikipedia](https://en.wikipedia.org/wiki/Vector_calculus_identities) — curl of a gradient: “It can be easily proved by expressing ∇ × (∇ φ) in a Cartesian coordinate system with Schwarz’s theorem (also called Clairaut’s theorem on equality of mixed partials).”; divergence of a curl: “The divergence of the curl of any continuously twice-differentiable vector field A is always zero,” and “This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.”; gradient: “the gradient is proportional to, and points in the direction of, the function’s most rapid (positive) change”; Laplacian: “When the Laplacian is equal to 0, the function is called a harmonic function.” - [Conservative vector field, Wikipedia](https://en.wikipedia.org/wiki/Conservative_vector_field) — “a conservative vector field is a vector field that is the gradient of some function”; irrotational: “if its curl is 0 everywhere in U”; and “the work W done in going around a simple closed loop C is 0.” ### 11.7 {#r-11.7} The inverse-distance potential satisfies ∇²(1/r)=0 for r≠0 and equals −4πδ at the origin in the distributional formulation. *Status: confirmed* - [Wolfram MathWorld, Laplacian](https://mathworld.wolfram.com/Laplacian.html) — derives that the Laplacian of 1/r vanishes away from the source and gives the −4π delta-function identity. ### 11.8 {#r-11.8} The divergence of the position field in three dimensions is 3 (n in n dimensions), so that the flux of r out of a closed surface is three times the volume inside, and the Laplacian of r²=x²+y²+z² is 6. As the radius R shrinks, the average of f over a sphere exceeds f at the center by ∇²f·R²/(2n) to leading order, R²∇²f/6 in three dimensions. *Status: confirmed* - [Wolfram MathWorld, Vector Derivative](https://mathworld.wolfram.com/VectorDerivative.html) — equation 22 gives ∇·r=3. - **Own computation** (the author's own work) — ∇²(x²+y²+z²)=2+2+2=6; in n dimensions the corresponding value is 2n. Likewise ∇·r=1+1+…+1=n in n dimensions, and by the divergence theorem ∮ r·dA = ∫ ∇·r dV = 3V in three dimensions. - [Laplace operator, Wikipedia](https://en.wikipedia.org/wiki/Laplace_operator) — “∇²f(x) = lim_{R→0} (2n/R²)(f_shell_R − f(x)),” where n is the dimension of the space and f_shell_R is the average of f over the surface of an n-sphere of radius R. ### 11.9 {#r-11.9} The flux of r̂/r² through a unit sphere is 4π, the solid angle of all directions, and Green’s line integral 1/2∮(x dy−y dx) around the unit circle is π. The solid angle of the whole sphere is 4π, and the flux of an inverse-square field through any closed surface is 4π times the enclosed source (for gravity, −4πGM). *Status: confirmed* - [Wolfram MathWorld, Stokes’ Theorem](https://mathworld.wolfram.com/StokesTheorem.html) — states the surface-flux/line-circulation relation underlying these integrals. - **Own computation** (the author's own work) — On S², r̂/r²·dA=sinθ dθ dφ, whose integral is 4π. Parametrizing the unit circle by x=cos t,y=sin t gives 1/2∫(x y’−y x’)dt=1/2∫_0^{2π}dt=π. - [Solid angle, Wikipedia](https://en.wikipedia.org/wiki/Solid_angle) — “The solid angle of a sphere measured from any point in its interior is 4π sr.” - [Gauss’s law for gravity, Wikipedia](https://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity) — “The gravitational flux through any closed surface is proportional to the enclosed mass”: ∯∂V g · dA = −4πGM. ### 11.10 {#r-11.10} The divergence theorem relates total outward flux through a closed surface to the volume integral of divergence; the historical attributions in the card are three special cases in Gauss’s paper of 1813 and Ostrogradsky’s general proof, presented in 1826 and published in 1831. Corrected 11 September 2026: this entry previously carried “Lagrange 1762,” which Katz’s history does not support — see the note at 11.25. Katz 1979 (see 11.25): “In three special cases it occurs in an 1813 paper of Gauss.” *Status: confirmed* - [Wikipedia, Divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem) — states the flux/divergence relation and documents Lagrange’s 1762 special case, Gauss’s 1813 cases, and Ostrogradsky’s 1826/1831 work. ### 11.11 {#r-11.11} Green’s essay was published in Nottingham in 1828, and the special line integral gives the area enclosed by a planar curve. Green’s theorem is the two-dimensional case of Stokes’ theorem, and ½∮(x dy − y dx) is the enclosed area; π is transcendental (Lindemann 1882). *Status: confirmed* - [Cambridge Core, Green’s and allied theorems: a historical sketch](https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/greens-and-allied-theorems-a-historical-sketch/158EF85DB56287D429B35222BC9730D4) — states that Green’s essay was published at Nottingham in 1828 and discusses the transformation on which the theorem depends. - [Green’s theorem, Wikipedia](https://en.wikipedia.org/wiki/Green%27s_theorem) — “It is the two-dimensional special case of Stokes’ theorem (surface in R³).”; area: “A = ∮_{C} x dy = −∮_{C} y dx = ½∮_{C}(−y dx + x dy).”; history: Green “stated a similar result in an 1828 paper”; “In 1846, Augustin-Louis Cauchy published a paper stating Green’s theorem as the penultimate sentence. A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his doctoral dissertation.” - [Pi, Wolfram MathWorld](https://mathworld.wolfram.com/Pi.html) — “π is also transcendental (Lindemann 1882).” - [J. Cufí and A. Reventós, A historical review of the Cauchy-Riemann equations and the Cauchy Theorem, arXiv:2311.16649](https://arxiv.org/pdf/2311.16649) — “Green’s formula (16) does not appear in Green’s paper”; “The first published proof of Green’s formula is due to Riemann in his 1851 dissertation.” - [n-sphere, Wikipedia](https://en.wikipedia.org/wiki/N-sphere) — “A unit 1-sphere is the unit circle in the Euclidean plane.” ### 11.12 {#r-11.12} Kelvin communicated the classical curl theorem to Stokes in 1850, Stokes used it in the 1854 Smith’s Prize examination, and the modern differential-form generalization is associated with Cartan after earlier work by Volterra, Goursat and Poincaré. *Status: confirmed* - [Wikipedia, Generalized Stokes theorem](https://en.wikipedia.org/wiki/Generalized_Stokes_theorem) — gives Kelvin’s 1850 letter, the 1854 exam question, and Cartan’s modern formulation following earlier generalizations; its note on the exam: “Maxwell took this examination and tied for first place with Edward John Routh.”; and “Stokes set the theorem as a question on the 1854 Smith’s Prize exam, which led to the result bearing his name.”; and “Stokes’ theorem was formulated in its modern form by Élie Cartan in 1945, following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.” ### 11.13 {#r-11.13} Maxwell published twenty scalar equations in 1865, while Heaviside and Gibbs-era vector notation condensed the system to the four equations now commonly taught; the equations imply electromagnetic waves with speed 1/√(μ₀ε₀). The identification of light as an electromagnetic wave is Maxwell’s own, made in the 1865 paper from his own twenty equations, and not a consequence of the later four-equation form; Heaviside and Gibbs supplied the notation, two decades after the prediction. Gradient, divergence and curl were first studied in the classical theory of electromagnetism. *Status: confirmed* - [Wikipedia, History of Maxwell’s equations](https://en.wikipedia.org/wiki/History_of_Maxwell%27s_equations) — documents Maxwell’s 1865 formulation, the twenty-equation presentation, and the 1884 vector condensation associated with Heaviside and Gibbs; it also records the electromagnetic-wave result. - [Nabla symbol, Wikipedia](https://en.wikipedia.org/wiki/Nabla_symbol) — “They were all originally studied in the context of the classical theory of electromagnetism, and contemporary university physics curricula typically treat the material using approximately the concepts and notation found in Gibbs and Wilson’s Vector Analysis.” ### 11.14 {#r-11.14} A divergence-free field that is twice continuously differentiable on all of ℝ³ and decays at least as fast as 1/‖x‖ is the curl of a vector potential. On a smaller region, whether a divergence-free field is a curl depends on the shape of the region, and the inverse-square field on punctured space is the chapter’s own obstruction. In the Aharonov–Bohm effect a charged particle is affected by the potential A where the magnetic field B is zero. *Status: confirmed* - [Wikipedia, Vector potential](https://en.wikipedia.org/wiki/Vector_potential) — gives the existence theorem for a solenoidal field on ℝ³ that is twice continuously differentiable and decreases at least as fast as 1/‖x‖ at infinity. - [MIT 18.022, When is a Vector Field the Curl of Another?](https://math.mit.edu/~djk/18_022/chapter08/section03.html) — states the result for a region R that is simply connected. See 11.18: simple connectivity is not by itself enough. - **Own computation** (the author's own work) — For r̂/r² on ℝ³∖{0}, the divergence is 0 everywhere the field is defined, but the flux through every enclosing sphere is 4π; the flux of a curl through a closed surface is 0, so the field is not a curl there. - [Aharonov–Bohm effect, Wikipedia](https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect) — “an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confined to a region in which both the magnetic field B and electric field E are zero.” ### 11.15 {#r-11.15} The nabla operator is the vector of partial derivatives, but the date of Hamilton’s first use of it is not the same in every source. The name comes from the Hellenistic Greek νάβλα for a Phoenician harp; the Nabla symbol page calls the often-claimed Hebrew derivation mistaken. MacTutor credits Tait’s Elementary Treatise on Quaternions of 1867 with establishing ∇ as the conventional symbol, and William Robertson Smith proposed the name to Tait in a letter of 10 November 1870. *Status: judgement* - [Wikipedia, Nabla symbol](https://en.wikipedia.org/wiki/Nabla_symbol) — says Hamilton introduced the operator in 1831, writing it as a rotated ◁, and that the name nabla was proposed by William Robertson Smith in a letter to Peter Guthrie Tait of 10 November 1870; its footnote: “Letter from Smith to Tait, 10 November 1870: My dear Sir, The name I propose for ∇ is, as you will remember, Nabla.” - [MacTutor, Earliest Uses of Symbols of Calculus](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/) — dates Hamilton’s use of the vector differential operator to the Royal Irish Academy meeting of 20 July 1846, published in 1847, and credits Tait’s Elementary Treatise on Quaternions of 1867 with establishing ∇ and the term nabla. Editorial note: Wikipedia dates Hamilton’s first use to 1831 and MacTutor to 1846, so we give no first-use date at all. The card names instead the two dates the sources agree on: Tait’s treatise of 1867 for the symbol, and Smith’s letter of 1870 for the name. ### 11.16 {#r-11.16} The exterior derivative satisfies d∘d=0, and in three dimensions this packages curl grad=0 and div curl=0; Stokes’ theorem relates ∫_M dω to ∫_{∂M}ω. Élie Cartan first described the exterior derivative in its present form in 1899. d² = 0 comes from the symmetry of second derivatives; its dual, ∂∂ = 0, is required of every chain complex in homology. *Status: confirmed* - [Wolfram MathWorld, Stokes’ Theorem](https://mathworld.wolfram.com/StokesTheorem.html) — identifies the differential-form form of Stokes’ theorem and its gradient/curl/divergence instances. - [Wikipedia, Exterior derivative](https://en.wikipedia.org/wiki/Exterior_derivative) — “The exterior derivative was first described in its current form by Élie Cartan in 1899”, and gives d(dα)=0 for a k-form α. - [Closed and exact differential forms, Wikipedia](https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms) — “Because d² = 0, every exact form is necessarily closed.”; “The implication from ‘exact’ to ‘closed’ is then a consequence of the symmetry of second derivatives, with respect to x and y.” - [Homology (mathematics), Wikipedia](https://en.wikipedia.org/wiki/Homology_(mathematics)) — “It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n, ∂ₙ ∘ ∂ₙ₊₁ = 0.” ### 11.17 {#r-11.17} Fermat’s method of maxima and minima, which locates an extremum by setting the rate of change to zero, circulated in a Latin treatise of about 1636, before there was a differential calculus to put it in. Fermat’s theorem puts unconstrained optima at stationary points, and Fermat and Lagrange found the calculus-based formulae for identifying optima. *Status: confirmed* - [Wikipedia, Fermat’s method of adequality](https://en.wikipedia.org/wiki/Adequality) — dates the treatise Methodus ad disquirendam maximam et minimam to about 1636 and describes Fermat’s method for maxima and minima before modern differential calculus. - [Mathematical optimization, Wikipedia](https://en.wikipedia.org/wiki/Mathematical_optimization) — “One of Fermat’s theorems states that optima of unconstrained problems are found at stationary points, where the first derivative or the gradient of the objective function is zero”; “Fermat and Lagrange found calculus-based formulae for identifying optima, while Newton and Gauss proposed iterative methods for moving towards an optimum.” ### 11.18 {#r-11.18} Simple connectivity is not sufficient for a divergence-free field to be a curl. ℝ³ with the origin removed is simply connected, and 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. *Status: confirmed* - [Wikipedia, Poincaré lemma](https://en.wikipedia.org/wiki/Poincar%C3%A9_lemma) — every closed p-form on a star-shaped open subset of ℝⁿ is exact, so the de Rham cohomology of such a region vanishes in every positive degree. - [Wikipedia, Vector potential](https://en.wikipedia.org/wiki/Vector_potential) — gives the existence theorem on ℝ³ and the restriction of the integral domain to a region. - **Own computation** (the author's own work) — Every loop in ℝ³∖{0} contracts, so the region is simply connected. The flux of r̂/r² through a sphere about the origin is 4π, while the flux of any curl through a closed surface is 0 by the divergence theorem and div curl = 0. So r̂/r² is not a curl there, and simple connectivity cannot be the sufficient condition. ### 11.19 {#r-11.19} The epigraph is Feynman’s: “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.” It is in the lecture on electromagnetism that opens Volume II, published in 1964. *Status: confirmed* - [Wikiquote, Richard Feynman](https://en.wikiquote.org/wiki/Richard_Feynman) — gives the passage — “the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics” — located in The Feynman Lectures on Physics, vol. II, lecture 1, section 1-6, p. 1-11. - [John D. Cook, What was the most important event of the 19th century?](https://www.johndcook.com/blog/2011/05/30/feynman-maxwell-equations/) — quotes the sentence in full, with the dashes, and attributes it to The Feynman Lectures on Physics, Volume 2. ### 11.20 {#r-11.20} Green, almost entirely self-taught, worked as a miller at the mill his father built in Nottingham. He printed his 1828 essay at his own expense and sold it by subscription 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 January 1845, showed the essay to Liouville and Sturm in Paris that summer, and had it reprinted between 1850 and 1854. *Status: confirmed* - [MacTutor, George Green](https://mathshistory.st-andrews.ac.uk/Biographies/Green/) — the essay was advertised in the Nottingham Review on 14 December 1827 and published in March 1828 by subscription at seven shillings and sixpence, with 51 subscribers, most of them members of the Nottingham Subscription Library; after Thomson took his degree in January 1845 his coach William Hopkins gave him three copies, and Thomson showed the work to Liouville and Sturm in Paris in the summer of 1845 and republished it in 1850–54. - [Wikipedia, George Green (mathematician)](https://en.wikipedia.org/wiki/George_Green_(mathematician)) — the essay was sold on a subscription basis to 51 people; Thomson had seen a reference to Green’s work in an 1833 paper by Robert Murphy but could find no bookseller who remembered it; the essay was not reprinted in Britain until 1871; “As a mathematician, he was almost entirely self-taught, receiving only about one year of formal schooling as a child, between the ages of eight and nine. Green then worked his father’s bakery from a young age, and later at the mill his father constructed in 1807, now known as Green’s Windmill. Whilst working as a miller, he became a member of Nottingham Subscription Library.” - [Cambridge Core, Green’s and allied theorems: a historical sketch](https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/greens-and-allied-theorems-a-historical-sketch/158EF85DB56287D429B35222BC9730D4) — states that Green’s essay was published at Nottingham in 1828 and was unknown to continental mathematicians until the reprints of 1850, 1852 and 1854. Editorial note: Two claims have to be separated here. That the essay went entirely unread is too strong: it had fifty-one subscribers, and Robert Murphy cited it in 1833. That Thomson was handed three copies in 1845 and carried the essay to Paris is documented by both biographies, so the card states it plainly rather than gesturing at reprints. ### 11.21 {#r-11.21} Laplace made the equation for the gravitational potential 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. Bibm@th dates the memoir to 1782; Rouse Ball says it was read in 1784. Siméon Denis Poisson showed in 1813 that inside the attracting matter the potential satisfies ∇²V = −4πρ rather than ∇²V = 0, so that the whole of the missing divergence sits where the mass or charge is. Sources differ on Poisson’s year: MacTutor and Rouse Ball give 1813, while Wikipedia’s Poisson’s equation says he published it in 1823. Corrections welcome. The Poisson equation for gravity follows from Gauss’s law in differential form. *Status: judgement* - [W. W. Rouse Ball, A Short Account of the History of Mathematics, Laplace](https://www.maths.tcd.ie/pub/HistMath/People/Laplace/RouseBall/RB_Laplace.html) — the memoir on the attraction of spheroids was read in 1784 and reprinted in the third volume of the Mécanique céleste; it develops the use of the potential, which satisfies ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z² = 0. - [MacTutor, Pierre-Simon Laplace](https://mathshistory.st-andrews.ac.uk/Biographies/Laplace/) — “although we now name this equation after Laplace, it was in fact known before the time of Laplace”. - [MacTutor, Siméon-Denis Poisson](https://mathshistory.st-andrews.ac.uk/Biographies/Poisson/) — “In 1813 Poisson studied the potential in the interior of attracting masses, producing results which would find application in electrostatics.” - [W. W. Rouse Ball, A Short Account of the History of Mathematics, Poisson](https://www.maths.tcd.ie/pub/HistMath/People/Poisson/RouseBall/RB_Poisson.html) — Poisson substituted ∇²V = −4πρ for Laplace’s ∇²V = 0, and this was first published in 1813. - [Laplacien, Bibm@th dictionnaire](https://www.bibmath.net/dico/index.php?action=affiche&quoi=.%2Fl%2Flaplacien.html) — gives 1782 and the polar coordinates: “La première apparition du laplacien apparait dans un mémoire de Laplace sur l’attraction des sphéroïdes en 1782, sous une forme compliquée due au système de coordonnées polaires qu’il utilisait.” - [Poisson’s equation, Wikipedia](https://en.wikipedia.org/wiki/Poisson%27s_equation) — gives 1823: “The equation is named after French mathematician and physicist Siméon Denis Poisson who published it in 1823.”; and “Gauss’s law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity.” Editorial note: The date of the spheroids memoir is not agreed. Rouse Ball says it was read in 1784; Wikipedia’s article on the equation dates Laplace’s study of its properties to 1786; the Mémoires volume the memoir belongs to is the one for 1782, printed in 1785. Ruled, 10 September 2026. The page names the volume rather than asserting a year: the memoir is the one in the Mémoires volume for 1782, which is checkable, and the bare date has been dropped from the chapter prose, where it carried no weight. The three accounts are not in conflict about a fact — an Académie volume is named for its session year while the memoir inside it is read and printed later — so recency cannot and should not separate them. The page claims nothing about who had the equation first, and nothing in the mathematics turns on the date. A later pass should not restore a bare 1782 to the prose: the volume year is not the year the memoir was read. ### 11.22 {#r-11.22} Gauss’s law for an inverse-square field is the first of the lemmas Carl Friedrich Gauss proved in his Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata, published in 1813 in Commentationes societatis regiae scientiarium Gottingensis recentiores vol. II, pp. 355–378; the text in the fifth volume of his collected works is a reprint, not a posthumous first appearance. What Gauss proves there is a special case; he came back to inverse-square forces in general later, in his Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte, Leipzig, 1840. That work’s title and year are carried by the Carl Friedrich Gauss line; no source read here says which of its theorems is the general flux law, so the book does not say that it is. *Status: confirmed* - [Wikipedia, Gauss’s law](https://en.wikipedia.org/wiki/Gauss%27s_law) — “The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1835, both in the context of the attraction of ellipsoids”, citing Gauss’s Werke, Band 5. The book no longer follows this page’s 1773 and 1835 (the Werke text is a reprint of the 1813 paper; see 11.25), but keeps the link as the record of the older account. The same page says Gauss’s law “is one of Maxwell’s equations” and that “Gauss’s law itself is essentially equivalent to Coulomb’s law, and Gauss’s law for gravity is essentially equivalent to Newton’s law of gravity, both of which are inverse-square laws.” - [Wikipedia, Divergence theorem](https://en.wikipedia.org/wiki/Divergence_theorem) — read as raw HTML: its history footnote gives the printing exactly — “C. F. Gauss (1813) ‘Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata,’ Commentationes societatis regiae scientiarium Gottingensis recentiores, 2: 355–378; Gauss considered a special case of the theorem” — and its running text says Gauss was “working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem”, that Lagrange “discovered the divergence theorem in 1762”, and that the first proof of the general theorem is Ostrogradsky’s, of 1826. - [Carl Friedrich Gauss, Wikipedia](https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss) — read as raw HTML: its list of works gives “1840: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte … Leipzig: Weidmannsche Buchhandlung, 1840,” translated there as General Theorems concerning the attractive and repulsive Forces acting in reciprocal Proportions of quadratic Distances. Editorial note: The fifth volume of Gauss’s Werke was first published in 1867 and reissued in 1877, and sources cite both years, so the card names the volume and not the year. Extended 17 September 2026. The entry previously asserted the 1813 printing while its only source said “followed by Carl Friedrich Gauss in 1835”; the Divergence theorem article gives the 1813 volume and pages outright, so the printing is settled and the entry no longer disputes its own evidence. Corrected later the same day: the Lagrange question is settled, and the evidence was in the book. r-11.25’s note records a deep-research pass of 11 September 2026 in which Katz 1979 was read in full and Lagrange’s memoirs were read in the original, and which concluded that “Lagrange 1773 is not a statement of Gauss’s law: the memoir was read end to end and contains no closed surface, no flux and no 4π; what it supplies is the change of variables Gauss’s proof runs on, which is a real priority for a method and not for the law.” The #flux card’s clause was briefly cut on 17 September 2026 for want of a source and has been restored, now citing 11.25, which the card had never cited. So the general-reference sentence that Lagrange “first formulated” the law is not followed here, and the reason is a reading of the primary document, not a preference between encyclopedias. ### 11.23 {#r-11.23} The fundamental theorem of calculus was reached by stages: James Gregory published the first statement and proof of a rudimentary form, Isaac Barrow proved a more generalized version, Isaac Newton stated the theorem explicitly in his tract on fluxions of October 1666 and completed the surrounding theory, and Gottfried Leibniz, working in Paris in the 1670s from quite different foundations, systematized the subject and introduced the notation still used, publishing his integral calculus in 1684 and 1686. *Status: judgement* - [MacTutor, The rise of calculus](https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/) — records that ‘Newton wrote a tract on fluxions in October 1666’ and that ‘this work contains the first clear statement of the Fundamental Theorem of the Calculus’; that ‘the idea that integral and derivative were inverses to each other were familiar to Barrow’ although ‘Barrow never explicitly stated the fundamental theorem of the calculus’; and that ‘On returning to Paris Leibniz did some very fine work on the calculus, thinking of the foundations very differently from Newton’, with ‘His results on the integral calculus were published in 1684 and 1686’. - [Wikipedia, Fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus) — the History section states that ‘The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by James Gregory (1638-1675)’, that ‘Isaac Barrow (1630-1677) proved a more generalized version of the theorem, while his student Isaac Newton (1642-1727) completed the development of the surrounding mathematical theory’, and that ‘Gottfried Leibniz (1646-1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today’. Editorial note: Added in the V14 prose pass. The chapter sentence ‘which Newton and Leibniz each reached in the late seventeenth century’ previously stood on 11.2, whose claim covers only derivative notation and carries no history of the theorem. Neither source here divides the credit the way that sentence does: Gregory and Barrow come before Newton, and the sources describe Newton as stating the theorem and Leibniz as systematizing the calculus rather than as two men each arriving at the same theorem. The wording is left for the author. ### 11.24 {#r-11.24} Newton and Leibniz were the parties to a priority dispute over who had first invented the calculus; the modern consensus is that they invented and described it independently in the seventeenth century. *Status: confirmed* - [Wikipedia, Leibniz-Newton calculus controversy](https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy) — opens ‘the calculus controversy was a priority dispute between mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus’, and states ‘Today, the consensus is that Leibniz and Newton independently invented and described calculus in Europe in the 17th century’. Editorial note: Added in the V14 prose pass to carry the chapter’s clause ‘the argument about whose calculus it was outlived them both’, which previously stood on 11.2. The source carries the dispute and the independence, but its narrative ends around 1716 and it does not state that the argument continued past the deaths of the two men. That half of the clause is still uncarried. ### 11.25 {#r-11.25} The divergence theorem carries at least six names and four dates in the reference literature — Lagrange 1762, Gauss 1813 (with further special cases in 1833 and 1839), Ostrogradsky 1826 for the first general proof, published 1831, and Green 1828 for special cases — and Gauss’s law has its own pair, Lagrange 1773 and Gauss 1835. The book gives Gauss 1813 for the special cases and Ostrogradsky 1826 for the general theorem, and for Gauss’s law it follows Gauss’s paper of 1813: Lagrange’s 1773 memoir contains no flux theorem, and the 1835 date rests on the Werke reprint of that paper (see 11.22). *Status: judgement* - [Wikipedia, Divergence theorem (History)](https://en.wikipedia.org/wiki/Divergence_theorem) — Lagrange ‘introduced the notion of surface integrals in 1760 and again in more general terms in 1811’ and ‘discovered the divergence theorem in 1762’; Gauss ‘proved special cases of the divergence theorem’ in 1813 while working on the gravitational attraction of an elliptical spheroid and ‘proved additional special cases in 1833 and 1839’; ‘it was Mikhail Ostrogradsky, who gave the first proof of the general theorem, in 1826, as part of his investigation of heat flow’; and ‘special cases were proven by George Green in 1828’ in the Essay on electricity and magnetism. - **Own reading** (the author's own work) — Lagrange’s 1773 memoir, read end to end for this book: it supplies the change of variables the divergence proof runs on, but has no closed surface, no flux and no 4π; the 1773 date on Gauss’s law is the literature’s attribution, not the memoir’s content. - [Encyclopedia of Mathematics, Ostrogradski formula](https://encyclopediaofmath.org/wiki/Ostrogradski_formula) — records that the identity is ‘often referred to as: Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula or Gauss-Green-Ostrogradski formula’, that the three-dimensional formula ‘is attributed to Gauss, who proved a particular case in 1813, and to Ostrogradski, who later generalized it to general dimension’ (1831, generalized 1838, both Mem. Acad. Sci. St Petersbourg), and that ‘the formula is already present in the works of Euler and other mathematicians of the 18th century’. - [MacTutor, Mikhail Ostrogradski](https://mathshistory.st-andrews.ac.uk/Biographies/Ostrogradski/) — Ostrogradski submitted Demonstration d’un theoreme du calcul integral to the Paris Academy on 13 February 1826, in which he ‘states and proves the general divergence theorem’; quoting Victor Katz, he ‘presented this theorem again in a paper in Paris on 6 August 1827, and finally in St Petersburg on 5 November 1828. The latter presentation was the only one published by Ostrogradski, appearing in 1831 in Note sur la Theorie de la Chaleur’; the page adds that Gauss proved special cases in 1833 and 1839 and that ‘the theorem is now often named after Gauss’. Editorial note: Added for Q11.3, ruled on by the author on 11 September 2026: the prose takes the earliest date, and the unclarity gets a card of its own rather than a silent choice. The card is #divattrib. DEEP RESEARCH DONE, 11 September 2026, and it changed the book. Katz 1979 was read in full and Lagrange’s two memoirs were read in the original. Lagrange 1762 is not a statement of the divergence theorem — it integrates by parts and then disposes of the boundary terms — and the credit traces to a footnote in a general-reference article rather than to any historian. Lagrange 1773 is not a statement of Gauss’s law: the memoir was read end to end and contains no closed surface, no flux and no 4π; what it supplies is the change of variables Gauss’s proof runs on, which is a real priority for a method and not for the law. Nothing supports 1835; the citation attached to that year in the article that prints it is the 1813 paper. And Gauss’s paper was not posthumous: it was published in 1813, and the text in Werke v. 5 is a reprint. All four claims have been removed from the prose and the cards. What remains genuinely contested is the divergence theorem itself — Gauss’s three special cases of 1813, Ostrogradsky’s general proof of 1826, Green’s special cases of 1828 — so the entry stays a judgement, and under D1 the prose prints the earliest it can source, ~1813, and names Gauss. ### 11.26 {#r-11.26} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: pi is transcendental by Lindemann’s theorem of 1882; Gibbs and Heaviside separated nabla from the quaternions in the 1880s. *Status: confirmed* - [Lindemann–Weierstrass theorem, Wikipedia](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem) — read raw: “Lindemann proved in 1882 that e^α is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below)..” - [Euclidean vector, Wikipedia](https://en.wikipedia.org/wiki/Euclidean_vector) — read raw: “The first half of Gibbs’s Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis..” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 11.31 {#r-11.31} The loop on an integral sign, ∮, marks an integral taken all the way round something closed: a line integral round a closed curve, or a surface integral over a closed surface. *Status: confirmed* - [Maxwell’s equations, Wikipedia](https://en.wikipedia.org/wiki/Maxwell%27s_equations) — read via the REST endpoint (revision 1375267474), in its key to the integral signs: the closed surface integral over ∂Ω “is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed,” and ∮ over ∂Σ “is a line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed..” - [MacTutor, Earliest Uses of Symbols of Calculus](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/) — under “Integration around a closed path”: “a use of the integral symbol with a circle in the middle by Arnold Sommerfeld (1868 - 1951) in 1917,” and E. B. Wilson “used a small circle below the standard integral symbol to denote integration around a closed curve.”