Joseph Fourier

How to use this course

Joseph Fourier asked a physical question: how does heat move through a solid body? To answer it he wrote down the heat equation, solved it by splitting temperatures into simple oscillating pieces, and claimed that almost any function could be built from sines and cosines. Analysts spent the next century making that claim precise, and the effort produced the modern notions of function, integral and convergence. Fourier analysis now runs through signal processing, quantum mechanics, probability and number theory.

He was also a revolutionary who was twice imprisoned, a member of Napoleon’s expedition to Egypt, a prefect for thirteen years, and the first person to treat the Earth’s surface temperature as a problem in the physics of radiation. This course follows both strands.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04 to 08 follow the order of the Théorie analytique de la chaleur: the equation, the lamina, the series, bodies of other shapes, and infinite bodies. Lessons 09 and 10 cover his geophysics and his work on equations.

Timeline

Auxerre and the Revolution, 1768–1794

Jean-Baptiste Joseph Fourier was born in Auxerre on 21 March 1768. His father was a tailor who had three children by a first marriage and twelve by a second; Joseph was the ninth of the twelve. His mother died when he was nine and his father the next year. Local patrons, among them the bishop of Auxerre, saw that the boy was gifted and found him a place in the town’s École Royale Militaire, which the Benedictines ran.

There he found mathematics. By about fourteen he had worked through Étienne Bézout’s Cours de mathématiques, and in 1783 he won a school prize for his study of Charles Bossut’s mechanics. A career as an artillery or engineering officer was the natural outlet for such a talent, but commissions in the scientific corps went to the sons of the nobility. Fourier turned to the Church instead. In 1787 he entered the Benedictine abbey of Saint-Benoît-sur-Loire as a novice, while continuing to work on algebra. In 1789 he sent a paper on the location of the roots of polynomial equations to the Académie Royale des Sciences in Paris; it was lost in the upheavals of that year, but the problem stayed with him for life (Lesson 10).

The Revolution

The Revolution closed the monasteries, and Fourier left Saint-Benoît in 1789 without taking his vows. In 1790 he returned to Auxerre to teach at his old school. He was drawn into politics, and in 1793 joined the local Revolutionary Committee. He later wrote that the ideal of equality had given him hope for a free government without kings or priests.

The Terror soon made that hope dangerous. Fourier defended members of a faction in Orléans who had fallen foul of the government’s envoys, and in July 1794 he was arrested. He was in prison when Maximilien Robespierre was guillotined on 28 July 1794, and was released in the weeks that followed. He was 26, had published nothing, and had already risked his life twice over.

Paris and Egypt, 1795–1801

Late in 1794 the Auxerre district chose Fourier as one of its students for the new École Normale in Paris, which opened in January 1795 to train teachers for the Republic. His professors were Joseph-Louis Lagrange, Pierre-Simon Laplace and Gaspard Monge. Fourier left a vivid judgement of Lagrange as a poor speaker but a profound teacher. The École Normale closed after a few months, but Fourier had been noticed.

He became an assistant at the École Centrale des Travaux Publics, soon renamed the École Polytechnique, working under Lagrange and Monge, and taught there from 1 September 1795. In the reaction against the former Jacobins that year he was arrested once more and soon released. In 1797 he succeeded Lagrange in the chair of analysis and mechanics. His surviving lecture notes from these years show a careful, conventional teacher of the analysis of the day.

The Egyptian expedition

In 1798 Monge recruited Fourier for the scientific commission that accompanied Napoleon Bonaparte’s invasion of Egypt. The fleet took Malta in June and Alexandria on 1 July. On 1 August Horatio Nelson destroyed the French fleet at Aboukir Bay, and the army and its savants were cut off from France.

In Cairo, Bonaparte founded the Institut d’Égypte on the model of the Institut de France. Fourier was one of the twelve members of its mathematics section and was elected its secretary, a post he held for the whole occupation. He also served as an administrator and diplomat, organised workshops, and studied the antiquities of Upper Egypt. Bonaparte left for France in 1799. Fourier stayed until the French capitulated in 1801 and returned with the remaining scholars. Egypt gave him two lasting things: the friendship and patronage of Bonaparte, and, by his own account, a conviction that desert heat was good for his health. In later life he kept his rooms very warm and wrapped himself heavily.

Prefect and physicist, 1802–1815

Fourier hoped to resume his professorship. In 1802 Bonaparte instead appointed him Prefect of Isère, the chief administrator of the department based in Grenoble. Fourier proved a capable prefect. He oversaw the draining of the marshes of Bourgoin, which had caused malaria, and the building of a road from Grenoble towards Turin over the Alps. He was made a baron of the Empire. He also took charge of the Description de l’Égypte, the vast record of the expedition, and wrote its long historical preface.

In Grenoble he encouraged the young Jean-François Champollion, whose interest in Egypt Fourier fed with his collection and his conversation, and whose elder brother Jacques-Joseph Champollion-Figeac worked with him. Champollion deciphered the hieroglyphs in 1822.

The heat memoirs

From about 1804 Fourier spent his free hours on the conduction of heat. Jean-Baptiste Biot had attempted the problem of a heated bar and run into difficulties with the idea of heat flowing between neighbouring particles. Fourier found the right equation, solved it, and on 21 December 1807 his memoir Sur la propagation de la chaleur dans les corps solides was read to the Institut. The examiners were Lagrange, Laplace, Monge and Sylvestre François Lacroix. Lagrange objected strongly to the use of trigonometric series to represent arbitrary functions, a point he had argued in the vibrating-string controversy decades earlier. Biot objected to the derivation of the equations. The memoir was not published.

The Institut then set the propagation of heat as the subject of its mathematics prize for 1811. Fourier submitted a revised memoir with new material on infinite bodies and the Fourier integral. A committee of Lagrange, Laplace, Étienne-Louis Malus, René Just Haüy and Adrien-Marie Legendre awarded him the prize, which was announced early in 1812, but their report complained that the analysis left something to be desired in generality and rigour. Fourier was stung. The prize memoir appeared in the Academy’s Mémoires only years later, after he had become the Academy’s secretary.

In 1814 Napoleon abdicated and Fourier stayed at his post under Louis XVIII. In March 1815 Napoleon returned from Elba and marched on Grenoble. Fourier left the city rather than receive him, but Napoleon, after a stormy meeting, appointed him Prefect of the Rhône at Lyon. Fourier resigned before Waterloo. The pension Napoleon granted him was never paid, and the restored monarchy regarded him with suspicion. At 47 he went to Paris without a post.

Flux, conduction and the heat equation

For the general reader. Heat flows from hot to cold, and faster when the temperature changes more steeply. Fourier turned this observation into a law, combined it with the conservation of energy, and obtained a single equation that governs the temperature at every point and every moment. The same equation describes diffusing dye, spreading populations and the prices of options.

The law of conduction

Let \(T(\mathbf x,t)\) be the temperature in a solid. Fourier postulated that heat flows down the temperature gradient with a flux proportional to its size. In modern vector notation:

Fourier’s advance over Biot was to treat the flux as a local quantity, defined at each point and depending only on the gradient there, rather than as a sum of actions between distant particles.

Derivation

Let \(\rho\) be the density and \(c\) the specific heat. The heat contained in a region \(V\) is \(\int_V\rho c\,T\,dV\). With no internal sources, it changes only by flux through the boundary:

The second equality is the divergence theorem. Since \(V\) is arbitrary, the integrands agree. For a homogeneous body this gives:

Boundary conditions

Fourier used three kinds, all still standard: prescribed temperature (\(T=T_b\) on the boundary), insulation (\(\partial T/\partial n=0\)), and surface exchange with a medium at temperature \(T_e\),

with outward normal \(n\) and exchange coefficient \(h\). The last is Newton’s law of cooling applied at the surface; Fourier called \(h\) the external conductivity. The ratio \(hL/k\) for a body of size \(L\) is now called the Biot number.

Uniqueness and the time scale

For prescribed boundary temperature \(T_b\), let \(w\) be the difference of two solutions with the same initial data. Then \(w=0\) on the boundary and

so \(\int w^2\) starts at 0 and cannot grow: the solution is unique. Scaling \(x=L\xi\), \(t=(L^2/\kappa)\tau\) removes all constants, so a body of size \(L\) cools on the time scale \(L^2/\kappa\). Doubling the size quadruples the time.

The rectangular lamina and separation of variables

For the general reader. Fourier’s first model problem was a long thin plate whose base is kept hot and whose sides are kept cold. He found simple solutions in which the temperature is a wave across the plate times a decay along it. Adding infinitely many of them matched the hot base exactly. The question of whether such sums can match anything at all became the subject of Fourier series.

The problem

A semi-infinite strip \(S=\{(x,y): x>0,\ |y|<\pi/2\}\) is kept at temperature 0 on its long sides \(y=\pm\pi/2\) and at 1 on its base \(x=0\). Find the steady temperature, bounded as \(x\to\infty\):

Separation of variables

Try \(v=X(x)Y(y)\). Then \(X''/X=-Y''/Y=\lambda\), a constant. The side conditions force \(Y=\cos(ny)\) with \(n\) odd, and boundedness forces \(X=e^{-nx}\). So each \(e^{-nx}\cos ny\), \(n=1,3,5,\dots\), is a solution vanishing on the sides. Fourier’s idea was to superpose them:

Fourier first found the \(a_n\) by an elaborate argument with infinitely many linear equations. He then saw the short route: multiply by \(\cos my\) and integrate, using orthogonality.

The series is a sum of smooth, even, periodic functions, yet it converges to a step. This was precisely what Lagrange found unacceptable. The steady temperature is therefore

The closed form follows from \(\sum_{j\ge0}\frac{(-1)^j z^{2j+1}}{2j+1}=\arctan z\) with \(z=e^{-x+iy}\): taking real parts of \(\arctan(e^{-x+iy})\) gives the formula above.

Fourier series

For the general reader. Fourier claimed that any function on an interval, even one given by different formulas on different pieces or with jumps, can be written as a sum of sines and cosines. The claim was too broad as he stated it, and too fruitful to abandon. Making it exact occupied Dirichlet, Riemann, Cantor and Lebesgue, and shaped modern analysis.

Coefficients

For an integrable \(2\pi\)-periodic \(f\), its Fourier series is

Leonhard Euler and others had computed such integrals before. Fourier’s contribution was to insist that the formulas make sense for any function whose integral can be drawn as an area, and to show what the resulting series do across dozens of examples. He drew the graphs of the partial sums and saw them approach functions with corners and jumps.

The ring

For a thin ring of circumference \(2\pi\), the heat equation is \(T_t=\kappa T_{xx}\) with \(T\) periodic in \(x\). If \(T(x,0)=f(x)\), then

High frequencies decay fastest; the ring tends to its mean temperature \(a_0/2\). This was the cleanest of Fourier’s applications and the model for all later eigenfunction expansions.

Convergence

Peter Gustav Lejeune Dirichlet, who knew Fourier in Paris, wrote the partial sum as an integral against the Dirichlet kernel and controlled it with the second mean-value theorem. Convergence is not automatic: Paul du Bois-Reymond found a continuous function whose Fourier series diverges at a point (1873), and Lennart Carleson proved in 1966 that the series of every square-integrable function converges almost everywhere.

The Gibbs phenomenon

Near a jump the partial sums overshoot. For the square wave \(\operatorname{sgn}(\sin x)=\frac4\pi\sum_{n\text{ odd}}\frac{\sin nx}{n}\), the maximum of the \(N\)-th partial sum tends to \(\frac2\pi\int_0^\pi\frac{\sin t}t\,dt\approx1.179\), an overshoot of about 9% of the jump. The overshoot narrows but never shrinks in height. Henry Wilbraham noticed it in 1848; Josiah Willard Gibbs explained it in 1899.

Rings, spheres and cylinders

For the general reader. For a plate or ring, the simple pieces are sines and cosines. For a sphere or a cylinder, Fourier found that the simple pieces are new functions, and their frequencies are roots of equations that cannot be solved in closed form. He argued that these roots are all real. The general theory of such expansions was built a few years later by Sturm and Liouville.

The sphere

A solid sphere of radius \(R\), initially heated symmetrically, cools by surface exchange into a medium at 0. With \(T=T(r,t)\), the heat equation is \(T_t=\kappa\,\frac1r(rT)_{rr}\). Setting \(u=rT\) gives \(u_t=\kappa u_{rr}\) with \(u(0,t)=0\). Separated solutions are \(T=e^{-\kappa\lambda^2t}\sin(\lambda r)/r\). The surface condition \(-kT_r=hT\) at \(r=R\) becomes

This equation has infinitely many positive roots \(\lambda_1<\lambda_2<\cdots\), and Fourier expanded the initial temperature in the functions \(\sin(\lambda_nr)/r\). The functions are not equally spaced in frequency, so this is not a Fourier series in the narrow sense. He argued from the physics that all roots must be real. The argument was disputed for years, and the general question was settled by the theory below.

The cylinder

For an infinitely long cylinder, separation in polar coordinates leads to

Fourier found this series in his heat memoirs and studied its roots, before Friedrich Bessel’s 1824 work on the functions now named after him.

Jacques Charles François Sturm, who was close to Fourier in his last years, and Joseph Liouville built this theory to settle exactly the questions Fourier had raised.

The Fourier integral and infinite bodies

For the general reader. On a bounded body only certain frequencies fit. On an infinite line every frequency is allowed, and the sum becomes an integral. Fourier’s 1811 memoir introduced this integral to treat heat spreading along an infinite bar. The result, the Fourier transform, is now among the most used tools in science.

From series to integral

Fourier wrote the result in a form close to

In complex notation, with \(\hat f(\xi)=\int_{\mathbb R}f(x)e^{-i\xi x}dx\):

Integration by parts gives \(\widehat{f'}(\xi)=i\xi\hat f(\xi)\). Differential equations with constant coefficients become algebraic.

The infinite bar

Solve \(T_t=\kappa T_{xx}\) on \(\mathbb R\) with \(T(x,0)=f(x)\). Transforming in \(x\): \(\hat T_t=-\kappa\xi^2\hat T\), so \(\hat T(\xi,t)=e^{-\kappa\xi^2t}\hat f(\xi)\). Inverting, a product becomes a convolution:

The heat kernel \(G\) is a Gaussian of width \(\sqrt{2\kappa t}\). Heat from a point source spreads as the square root of time, and its profile is the normal distribution that Laplace had found in probability. The link between diffusion and random walks, made exact by Louis Bachelier and Albert Einstein, starts here.

The temperature of the Earth and the greenhouse effect

For the general reader. Fourier asked why the Earth has the temperature it has. He separated three sources of heat: the Sun, the planet’s own interior, and the cold of space. He concluded that the interior contributes almost nothing at the surface, and that the atmosphere keeps the surface warmer by letting sunlight in while hindering the escape of the invisible heat the ground gives off. This is the first statement of what we now call the greenhouse effect.

Temperature waves in the ground

Let the surface temperature vary periodically, \(T(0,t)=\bar T+A\cos\omega t\). The bounded solution of \(T_t=\kappa T_{zz}\) for depth \(z>0\) is

Fourier used this to explain why deep cellars keep a nearly constant temperature, and why they are warmest when the surface is coolest.

Interior heat and the age of the Earth

Temperatures in mines rise with depth, so heat flows out of the interior. Fourier showed that this flux, about a tenth of a watt per square metre in modern units, warms the surface by only a small fraction of a degree. He modelled the Earth as a half-space cooling from a uniform initial temperature \(T_0\), surface held at 0. The solution is \(T=T_0\operatorname{erf}\big(z/(2\sqrt{\kappa t})\big)\), with surface gradient

Fourier noted that measured gradients imply a very long cooling time. In 1862 William Thomson (later Lord Kelvin) used exactly this formula to estimate the age of the Earth. Radioactive heating and mantle convection, unknown to both, make the real age far larger.

The atmosphere as a blanket

In 1824 Fourier published Remarques générales sur les températures du globe terrestre et des espaces planétaires in the Annales de chimie et de physique, with a longer version in the Academy’s Mémoires in 1827. He compared the atmosphere to the glass of Horace-Bénédict de Saussure’s heliothermometer, a box with glass panes that grew far hotter in sunlight than the outside air. Light heat (sunlight) passes through the air easily; dark heat (what we call infrared) radiated by the ground passes less easily. The surface therefore settles at a higher temperature than it would without an atmosphere. He also argued that space itself has a temperature, which he took to be rather cold but not absolute zero, and that the Earth’s temperature results from a balance between these influences.

Fourier had no theory of radiation to make this quantitative. John Tyndall measured the infrared absorption of water vapour and carbon dioxide in 1859, and Svante Arrhenius calculated the warming from doubled carbon dioxide in 1896. The simplest modern version follows.

Equations, inequalities, dimensions

For the general reader. Fourier’s first mathematical interest, as a novice at Saint-Benoît, was where the roots of a polynomial lie. He returned to it at the end of his life in a book left unfinished at his death. Along the way he described a method for solving systems of inequalities that anticipates linear programming, and he introduced the systematic use of physical dimensions.

Counting real roots

For a real polynomial \(p\) of degree \(n\), let \(V(x)\) be the number of sign changes in the sequence \(p(x),p'(x),\dots,p^{(n)}(x)\), ignoring zeros.

Ferdinand Budan published an equivalent result in 1807 and 1811; Fourier said he had taught it at the École Polytechnique in the 1790s, and published his version in 1820. Taking \(a=0\), \(b\to\infty\) recovers René Descartes’s rule of signs. The bound may overcount. In 1829 Sturm found a sequence that gives the exact count, and credited Fourier’s work as his starting point.

Linear inequalities

The synopsis of Fourier’s Analyse des équations déterminées and notes he published in the 1820s describe how to decide whether a system of linear inequalities has a solution. To eliminate a variable \(x\), write each inequality as a lower bound \(x\ge L_i\) or an upper bound \(x\le U_j\) (or neither) and replace them all by the pairwise conditions \(L_i\le U_j\). The method was rediscovered by Theodore Motzkin in 1936 and is now called Fourier–Motzkin elimination. Fourier also sketched a descent method for minimising the largest error of an overdetermined linear system, an early idea in what became linear programming.

Dimensions

In the Théorie analytique Fourier assigned to each quantity exponents of dimension in length, time and temperature, and observed that every term of a valid equation must have the same exponents. Conductivity \(k\), for example, has the dimensions of heat per unit time, per unit length, per degree. This is the origin of dimensional analysis.

Paris and the Academy, 1815–1830

In Paris after Waterloo, Fourier had no income. A former student, now Prefect of the Seine, made him director of the department’s statistical bureau. Fourier edited its statistical reports on Paris and wrote memoirs on the errors of averages and on the reliability of results drawn from data.

The Académie des Sciences elected him in 1816, but Louis XVIII refused to confirm a man who had served Napoleon during the Hundred Days. He was elected again in May 1817 and this time confirmed. In 1822, after the death of Jean-Baptiste Delambre, he was elected permanent secretary for the mathematical sciences, defeating Biot. The same year the Academy published the Théorie analytique de la chaleur, a revised and expanded version of his prize memoir. He became a Fellow of the Royal Society in 1823 and entered the Académie française in 1827.

Quarrels and protégés

Siméon Denis Poisson had his own theory of heat, built on molecular actions, and published solutions to problems Fourier considered his. Fourier answered sharply in print and in unpublished notes, and also contested Biot’s claims to priority. In 1823 he collaborated with the Danish physicist Hans Christian Ørsted on thermoelectric circuits.

As secretary he used his position to help young mathematicians. He supported Dirichlet, who came to Paris in 1822 and whose 1829 convergence theorem answered Fourier’s critics. He encouraged Sturm, whose root theorem grew from Fourier’s work on equations, and Claude-Louis Navier, his friend and later editor. He received the manuscript of Évariste Galois’s 1830 prize memoir as secretary; it was lost after his death and never judged.

Death

Fourier’s health declined in his last years. He suffered from what was probably heart disease and breathlessness, and he kept his rooms overheated. On 4 May 1830 he fell on the stairs at home. He died in Paris on 16 May 1830, aged 62, and was buried in the cemetery of Père Lachaise under a monument with Egyptian motifs. Navier published his unfinished book on equations in 1831.

Legacy

Fourier’s critics were right that his arguments were not rigorous, and he was right that his methods worked. Resolving the tension produced much of nineteenth-century analysis. Dirichlet’s convergence theorem required a precise notion of function. Bernhard Riemann’s 1854 habilitation, on trigonometric series, introduced the Riemann integral. Georg Cantor’s study of sets of uniqueness for trigonometric series led him to infinite sets and ordinals. Henri Lebesgue’s integral (1902) made Parseval’s identity and \(L^2\) theory clean.

• Physics and engineering: the heat equation, diffusion, conduction, and every method that expands a solution in eigenfunctions, from quantum mechanics to vibration analysis.
• Mathematics: harmonic analysis on groups, spectral theory, distributions and the fast Fourier transform (Cooley and Tukey, 1965).
• Climate science: the first physical account of a planet’s temperature as a radiative balance, the starting point for Tyndall and Arrhenius.
• Optimisation: Fourier–Motzkin elimination and early ideas of linear programming.
• Honours: his name is one of the 72 engraved on the Eiffel Tower. The university in Grenoble, now part of Université Grenoble Alpes, bore his name, and its mathematics institute, the Institut Fourier, publishes the Annales de l’Institut Fourier.

William Thomson read the Théorie analytique as a teenager and built his first papers on it, defending Fourier against a critic in print at sixteen. Few books have so directly shaped both physics and pure mathematics.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Jean Baptiste Joseph Fourier, MacTutor History of Mathematics.
2. J. R. Ravetz and I. Grattan-Guinness, Fourier, Jean Baptiste Joseph, Dictionary of Scientific Biography.
3. Joseph Fourier, Wikipedia; Joseph Fourier, Wikipédia (French).
4. J. Herivel, Joseph Fourier: The Man and the Physicist, Oxford, 1975.
5. I. Grattan-Guinness (with J. R. Ravetz), Joseph Fourier 1768–1830, MIT Press, 1972. Includes the 1807 memoir.
6. J. Dhombres and J.-B. Robert, Fourier, créateur de la physique mathématique, Belin, 1998.

Mathematics and physics

1. J. Fourier, Théorie analytique de la chaleur, Paris, 1822. English translation by A. Freeman, The Analytical Theory of Heat, Cambridge, 1878.
2. J. Fourier, Remarques générales sur les températures du globe terrestre et des espaces planétaires, Annales de chimie et de physique 27 (1824).
3. J. Fourier, Analyse des équations déterminées, ed. C.-L. Navier, Paris, 1831.
4. P. G. L. Dirichlet, Sur la convergence des séries trigonométriques, J. reine angew. Math. 4 (1829).
5. W. Thomson, On the secular cooling of the earth, Trans. Roy. Soc. Edinburgh 23 (1862).
6. J. R. Fleming, Joseph Fourier, the “greenhouse effect”, and the quest for a universal theory of terrestrial temperatures, Endeavour 23 (1999).
7. R. T. Pierrehumbert, Infrared radiation and planetary temperature, Physics Today 64 (2011).
8. T. W. Körner, Fourier Analysis, Cambridge, 1988.
9. E. M. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton, 2003.