But what is a Fourier series? From heat flow to circle drawings | DE4

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Jean-Baptiste Joseph Fourier (/ˈfʊrieɪ, -iər/;[1]French: [fuʁje]; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations. The Fourier transform, and Fourier’s law of conduction is also named in his honor. Fourier is also generally credited with the discovery of the greenhouse effect.[2]

The Analytic Theory of Heat

In 1822 Fourier published his work on heat flow in Théorie analytique de la Chaleur (The Analytical Theory of Heat),[7] in which he based his reasoning on Newton’s law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the minimal difference of their temperatures. This book was translated,[8] with editorial ‘corrections,’ [9] into English 56 years later by Freeman (1878).[10] The book was also edited, with many editorial corrections, by Darboux and republished in French in 1888.[9]

There were three critical contributions in this work, one purely mathematical, two primarily physical. In mathematics, Fourier claimed that any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable. Though this result is not correct without additional conditions, Fourier’s observation that some discontinuous functions are the sum of infinite series was a breakthrough. The question of determining when a Fourier series converges has been fundamental for centuries. Joseph-Louis Lagrange had given particular cases of this (false) theorem and had implied that the method was general, but he had not pursued the subject. Peter Gustav Lejeune Dirichlet was the first to give a satisfactory demonstration of it with some restrictive conditions. This work provides the foundation for what is today known as the Fourier transform.

One crucial physical contribution in the book was the concept of dimensional homogeneity in equations; i.e., an equation can be formally correct only if the dimensions match on either side of the equality; Fourier made essential contributions dimensional analysis.[11] The other physical gift was Fourier’s proposal of his partial differential equation for conductive diffusion of heat. This equation is now taught to every student of mathematical physics.

Fourier was placed in a tricky position in 1814, when Napoleon abdicated and set out for Elba with every likelihood of passing southward through Grenoble, on what has come to be known today as the Route Napoleon. To greet his old master, would jeopardize his standing with the new king, Louis XVIII, who, in any case, might not look favorably on former associates and appointees of the departing emperor. Fourier influenced the choice of a changed route and kept his job.

But the next year, Napoleon reappeared in France, marching north through Grenoble, where he fired Fourier, who had made himself scarce. Nevertheless, three days later, Fourier was appointed Prefect of the Rhône at Lyons, thus surviving two changes of régime. Of course, only 100 days elapsed before the king was back in control, and Napoleon was on his way to the south Atlantic, never to return.

Fourier’s days in the provincial government then ended. He moved to Paris to enter a life of science and scientific administration, being elected to the Académie des Sciences in 1817, to the position of permanent secretary in 1823, and the Académie Française in 1826.

Fourier never married. But, among his close friends was the first great female applied mathematician, Sophie Germain. They corresponded for years and died only nine months apart.

Discovery of the greenhouse effect

In the 1820s, Fourier calculated that an object the size of the Earth, and at its distance from the Sun, should be considerably colder than the planet is if warmed by only the effects of incoming solar radiation. He examined various possible sources of the additional observed heat in articles published in 1824[13] and 1827.[14] While he ultimately suggested that interstellar radiation might be responsible for a large portion of the additional warmth, Fourier’s consideration of the possibility that the Earth’s atmosphere might act as an insulator of some kind is widely recognized as the first proposal of what is now known as the greenhouse effect,[15] although Fourier never called it that.[16][17]

In his articles, Fourier referred to an experiment by de Saussure, who lined a vase with blackened cork. Into the cork, he inserted several panes of transparent glass, separated by intervals of air. Midday sunlight was allowed to enter at the top of the vase through the glass panes. The temperature became more elevated in the more interior compartments of this device. Fourier concluded that gases in the atmosphere could form a stable barrier like the glass panes.[14] This conclusion may have contributed to the later use of the metaphor of the “greenhouse effect” to refer to the processes that determine atmospheric temperatures.[18] Fourier noted that the actual mechanisms that determine the warmth of the atmosphere included convection, which was not present in de Saussure’s experimental device.


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