Jean Baptiste Joseph Fourier(1768-1830) - French mathematician and physicist, foreign honorary member of the St. Petersburg Academy of Sciences (1829). Works on algebra, differential equations and mathematical physics. His “Analytical Theory of Heat” (1822) was the starting point in the creation of the theory of trigonometric series (Fourier series).
The first works of J. Fourier relate to algebra. In lectures of 1796, he presented a theorem on the number of real roots of an algebraic equation lying between given boundaries (published in 1820), named after him; a complete solution to the question of the number of real roots of an algebraic equation was obtained in 1829 by J. S. F. Sturm.
In 1818, Fourier investigated the question of the conditions for the applicability of the method of numerical solution of equations developed by Isaac Newton, not knowing about similar results obtained in 1768 by the French mathematician J. R. Murail. The result of Fourier's work on numerical methods for solving equations is “Analysis of Definite Equations,” published posthumously in 1831.
Jean Fourier's main field of study was mathematical physics. In 1807 and 1811, he presented his first discoveries on the theory of heat propagation in solids to the Paris Academy of Sciences, and in 1822 he published the work “Analytical Theory of Heat,” which played a major role in the subsequent history of mathematics. In it, Fourier derived the differential equation of heat conduction and developed the ideas outlined earlier by Daniel Bernoulli, developed a method for separating variables (Fourier method) to solve the heat equation under certain given boundary conditions, which he applied to a number of special cases (cube, cylinder, etc.). This method is based on the representation of functions by trigonometric Fourier series, which, although sometimes considered earlier, became an effective and important tool of mathematical physics only with Fourier. The method of separation of variables was further developed in the works of S. Poisson, Mikhail Vasilyevich Ostrogradsky and other mathematicians of the 19th century.
The “Analytical Theory of Heat” was the starting point for the creation of the theory of trigonometric series and the development of some general problems of mathematical analysis. Fourier gave the first examples of the expansion into trigonometric Fourier series of functions that are specified in different areas by various analytical expressions. Thus, he made an important contribution to the resolution of the famous dispute about the concept of function, in which the greatest mathematicians of the 18th century participated. His attempt to prove the possibility of expanding any arbitrary function into a trigonometric Fourier series was unsuccessful, but marked the beginning of a large series of studies devoted to the problem of the representability of functions by trigonometric series (P. Dirichlet, Nikolai Ivanovich Lobachevsky, B. Riemann, etc.). The emergence of set theory and the theory of functions of a real variable was largely associated with these studies.
Chronicle:
Jean Fourier was born in Auxerre into the family of a tailor. At the age of 9, he lost both parents. The orphan was sent to a military school at a Benedictine monastery.
In 1789 he came to Paris to present a work on the numerical solution of equations of any degree, but it was lost during the revolution. The revolution came before he could decide what to become - a monk, a military man or a mathematician. Fourier returned to Auxerre and began teaching at the school where he had previously studied.
Jean Fourier actively contributed to the new government, and he was invited to the Normal School, organized by the Convention for the training of teachers (1794). The school was soon closed, but he managed to attract the attention of prominent scientists (Joseph Louis Lagrange, Pierre Simon Laplace and mathematician and engineer Gaspard Monge).
In 1795-1798, Fourier taught at the Polytechnic School. According to listeners, Fourier's lectures were brilliant.
1796: In his lectures, Fourier presents a theorem on the number of real roots in a given interval (published 1820). Later, his results were generalized by the Swiss Jacques Char-François Sturm (who worked in Paris) and the mathematician Augustin Louis Cauchy.
1798: Napoleon Bonaparte takes Fourier, Monge and Berthollet on his Egyptian campaign, as part of the Legion of Culture.
1801: Fourier returns to France and is appointed prefect of the Isère department. He is engaged in draining swamps. Writes "Mathematical Theory of Heat".
1808: Jean Fourier receives the title of baron from Napoleon and is awarded the Legion of Honor.
1812: Fourier wins the Academy's Grand Prize for his analytical theory of heat conduction, despite weak evidence. However, full rigor was achieved only in Hilbert's era.
He used his methods (Fourier series and integrals) in the theory of heat propagation. But they soon became an extremely powerful tool for the mathematical study of a wide variety of problems - especially where there are waves and oscillations. And this circle is extremely wide - astronomy, acoustics, tidal theory, radio engineering, etc.
1815: During the Hundred Days, Jean Fourier defects to the Emperor. After the Restoration, he was removed from the post of prefect and lived in poverty. Returns to Paris, where he worked for some time as director of the Bureau of Statistics.
1817: The Academy, despite pressure from the Bourbons, elects Jean Fourier as a member (the first attempt in 1816 failed, King Louis XVIII canceled the election). Fourier became one of the most influential academicians, and in 1822 he was elected secretary for life.
1822: The final classic treatise, “The Mathematical Theory of Heat” (Th?orie analytique de la chaleur), is published. Lord Kelvin called this work “the great mathematical poem”.
Fourier's law is the law of thermal conductivity of isotropic media established by a physicist in 1822, connecting the temperature gradient in the medium with the heat flux density.
Thermal conductivity is the transfer of energy from more heated areas of the body to less heated ones as a result of thermal movement and the interaction of its constituent particles. Leads to equalization of body temperature. Typically, the amount of energy transferred, defined as heat flux density, is proportional to the temperature gradient (Fourier's law). The proportionality coefficient is called the thermal conductivity coefficient.
Fourier series is a trigonometric series, the coefficient of which for a function f(x) given on the interval [ - ¶,¶] is calculated using the Euler - Fourier formulas:
Where k=1,2,...
Partial sums of the Fourier series are an important apparatus for approximate representation of the function f(x). Fourier series were widely used in the works of J. Fourier and other scientists.
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In 1794 he entered the Normal School, organized by the Convention for the training of teachers. Soon the school was closed, but he managed to attract the attention of prominent scientists (Lagrange, Laplace and Monge). In 1795 - 1798 he taught at the Polytechnic School.
Participated together with other scientists in Napoleon's Egyptian campaign. He was the secretary of the Cairo Institute, founded by Napoleon. After the victory of England, in 1802 he was appointed prefect of the Isère department with headquarters in Grenoble, where he continued his scientific research in algebra, and worked actively in a new field of physics - the theory of heat. In 1808, Fourier received the title of baron and was awarded the Legion of Honor.
After Napoleon's defeat at Waterloo and the end of the "hundred days", he was removed from the post of prefect and moved to Paris. Here he worked for some time as director of the Bureau of Statistics, and thanks to the experience gained in Egypt, he raised this business to a height. In 1816, the Paris Academy of Sciences elected him as a member, but King Louis XVIII canceled the election. In 1816, the Academy of Sciences again elected him as a member, but this time the election was confirmed. Fourier became one of the most influential academicians and in 1822 he was elected secretary for life. In the same year he published the Analytical Theory of Heat (Théorie analytique de la chaleur). He died on May 16, 1830 in Paris.
Scientific achievements
He proved a theorem about the number of real roots of an algebraic equation lying between given limits (Fourier's theorem 1796).
He studied, independently of J. Muraile, the question of the conditions for the applicability of the method of numerical solution of equations developed by Isaac Newton (1818).
Monograph “Analytical Theory of Heat”, in which the derivation of the heat conduction equation in a solid body was given, and the development of methods for its integration under various boundary conditions. The Fourier method consisted of representing functions in the form of trigonometric series (Fourier series).
I found a formula for representing a function using an integral, which plays an important role in modern mathematics.
He proved that any arbitrarily drawn line, composed of segments of arcs of different curves, can be represented by a single analytical expression.
In 1823, independently of Oersted, he discovered the thermoelectric effect, showed that it has the property of superposition, and created a thermoelectric element.
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France
Jean Baptiste Joseph Fourier(fr. Jean Baptiste Joseph Fourier; March 21, Auxerre, France - May 16, Paris), French mathematician and physicist.
Biography
early years
Jean Baptiste Joseph Fourier was the 12th of 15 children in a tailor's family (the ninth in his father's second marriage). His father, Joseph Fourier, came from a family of shopkeepers from the small town of Lorient. In the 16th and 17th centuries, Pierre Fourier, great-uncle of Jean Fourier, was a prominent figure in the Counter Reformation in the city. His mother, Edmie, died in 1777 when Fourier was nine years old, and his father died the same year. According to other sources, Fourier became an orphan at the age of eight.
At his first school, which was led by a church musician, Fourier showed success in studying French and Latin. At the age of 12, with the assistance of the Bishop of Auxerre, Fourier was sent to a military school at a Benedictine monastery. By the age of 13, Joseph became interested in mathematics, and at the age of 14 he mastered Bézout's six-volume Course of Mathematics. At the same time, he began collecting candle stubs in the school building so that he could study at night. In 1782-1783, Fourier received many prizes in rhetoric, mathematics, mechanics and singing. The long illness that followed may have been due to these intense activities.
At the age of 17, he dreamed of a military career and wanted to become an artilleryman or military engineer. Despite the support of school teachers and inspectors, Fourier was refused due to his humble origins. In 1787, Fourier entered the Benedictine Abbey of St. Benoit-sur-Loire, where he intended to be ordained. However, the young man doubted his choice, submitted documents to Montucla Paris, left the abbey in 1789 and went to the capital. In Paris, at the Royal Academy of Sciences, Fourier presented work on the numerical solution of equations of any degree.
During the Great Revolution
The revolution came before he could decide what to become - a monk, a military man or a mathematician. The revolutionary decree of October 1789 abolished religious vows, and soon the property of the church and monastic orders was confiscated. Fourier returned to Auxerre and began teaching mathematics, rhetoric, history and philosophy at the school from which he himself graduated. The Commissioner, who visited the school in October 1792, noted the liberal atmosphere of the classes and was dissatisfied only with the small number of Latin classes, which, at the request of the parents, gave way to mathematics classes
Until February 1793, Fourier was not involved in politics, despite the fact that the most militant provincial branch of the Jacobist party was located in Auxerre. In 1793, a heated debate took place in Auxerre on the principles of separating people from the region at the request of the Convention. Fourier spoke at this debate and proposed a plan which was ultimately supported. In March 1793, Fourier received an offer to join the Comite de Surveillance, which he accepted. In September of the same year, the committee that dealt with the affairs of travelers became part of the revolutionary terror and was obliged to arrest supporters of tyranny or federalism and enemies of freedom. Fourier, who did not want to participate in this, submitted a written resignation from the committee, which was rejected
On committee affairs he went to the Loiret department. While passing by Orleans, he became involved in a local conflict, speaking out in defense of the heads of several local families, when a representative of the Convention made many arrests and intended to use a mobile guillotine. As a result, on October 29, 1793, his powers were revoked with the impossibility of obtaining them in the future, and Fourier returned to Auxerre in fear, where he continued to be a member of the local party branch and teach at school. Moreover, in June 1794 he became president of the revolutionary committee in Auxerre. After this, Fourier went to Paris for a meeting with Robespierre, which was not successful, since on July 4, immediately upon returning to Auxerre, he was arrested. He was already expecting the guillotine when, as a result of the coup of 9 Thermidor, Robespierre was arrested and executed, after which Fourier was released
On October 30, 1794, by decree of the Convention in Paris, the Normal School was organized, in which 1,500 students were trained with money from the republic, who were to become school teachers. Students were nominated from various districts, in particular, since Auxerre nominated its candidate while Fourier was in prison, he was nominated by the neighboring district of Saint-Florentine and entered the school after confirmation from Auxerre. Such outstanding scientists as Lagrange, Laplace, Monge, Bertholet taught at the school. Classes began on January 20, 1795, but already in May 1795 the school ceased to exist.
At the same time, Fourier's opponents wrote a letter to the École Normale arguing that it was impossible to prepare teachers for children from those candidates who were chosen under Robespierre, in particular Fourier himself. In May 1795, two orders came to Auxerre: on May 12, to disarm the participants in the terror, including Fourier, and on May 30, to take those who refused into custody. By that time, Fourier had received a position at the École Polytechnique, which at that time had a different name. He tried to resist, abandoned his position and wrote a letter to the municipality of Auxerre, but on June 7 he was captured and sent to prison. From prison, he wrote many letters in his defense, claiming in particular that under Robespierre he was imprisoned and he owed his life and freedom to the coup of 9 Thermidor. In August 1795, for an unknown reason, Fourier was released. His release is associated with the changed political climate in the country, or with the possible intercession of Lagrange and Monge.
Egyptian campaign
In Grenoble
Bust of Fourier in Grenoble
Fourier returned to France in 1801 and was reinstated as professor at the École Polytechnique. However, Napoleon offered him the post of prefect of the Isère department, and Fourier could not refuse the offer and went to Grenoble. Fourier's main achievements in office were leading the draining of the marshes in Bourgoin, as well as the construction of a new road connecting Grenoble with Turin. At the same time, Fourier was working on the collection en:Description de l"Égypte. In addition to the selection of material, he wrote a historical reference to Ancient Egypt. The collection began to be published in 1810, after Napoleon made a number of changes to it (in the second edition the collection was published with original text).
In 1809, Fourier received the title of baron from Napoleon and was awarded the Legion of Honor.
In 1812, Napoleon was defeated and went into exile on Elba. His route was supposed to pass through Grenoble, but Fourier sent a note that the city might be unsafe. When Napoleon left Elba and went with his army through Grenoble, Fourier left the city in a hurry, which displeased Napoleon. Fourier was later able to gain the favor of the Emperor, who appointed him Prefect of the Rhône. However, Fourier soon left his post. On June 10, 1815, Napoleon awarded Fourier a pension in the amount of 6 thousand francs, but Fourier never received it, since Napoleon was defeated on July 1. After this, Fourier returned to Paris, where he worked for some time as director of the Bureau of Statistics, and in 1817 became a member of the Academy.
Later years
Thanks to his work in Egyptology, Fourier also became a member of the Académie Française and the Académie de Médecine in 1826.
In 1804, while in Grenoble, Fourier began work on the theory of heat propagation in solids. By 1807, he prepared a report “On the propagation of heat in a solid,” which he presented on December 21 of that year in Paris. The report received a very controversial assessment. Lagrange and Laplace could not accept the fact that Fourier expanded functions into trigonometric series, which were later named after him. Further explanations from Fourier also could not shake their point of view. In addition, Biot opposed Fourier's equation for the propagation of heat. Fourier in his work did not refer to a similar work by Biot, published by him in 1804. Laplace and later Poisson agreed with Biot. Later, in 1812, the analytical theory of thermal conductivity presented by Fourier received the Grand Prize of the Academy. However, full rigor was achieved only in the era of Hilbert.
In 1818, Fourier was preoccupied with the question of the conditions for the applicability of the method of numerical solution of equations developed by Newton. Similar results had already been obtained in 1768 by Murail. The results of this work were published only in 1831, after the death of the scientist.
In 1817, Fourier was elected a member of the Academy of Sciences, despite pressure from the Bourbons. The first attempt in 1816 failed, King Louis XVIII canceled the election. In 1822, after d'Alembert's death, he was able to take the post of secretary of the mathematical section. Soon after this, his work "The Analytical Theory of Heat" ("Théorie analytique de la chaleur") was published, which Lord Kelvin called "The Great Mathematical Poem". At this time, Fourier withdrew from mathematical research and was more occupied with publishing his work in both pure and applied mathematics. His theory of heat was still controversial, with Biot claiming primacy on the matter, and Poisson criticizing Fourier's mathematical approach and developing an alternative theory.
Teaching work
While studying at the Normal School, already an experienced teacher, Fourier assessed his teachers and their manner of delivering lectures. He noticed Lagrange's chaotic approach to presentation, as well as his errors in sentences, which Fourier considered to be a consequence of his Italian roots, while calling the latter an extraordinary person. He called Laplace's lectures accurate, but very fast and uninteresting. Monge's lectures, according to Fourier, were neat and understandable, delivered in a loud voice. He believed that Berthollet's lectures on chemistry could only be understood by someone who already knew the subject, since he spoke with difficulty, hesitated and repeated himself a lot.
While recruiting students at the Ecole Polytechnique, Fourier believed that talent was more important than diligence. One of Fourier's students was Poisson, who replaced him at school during the Egyptian campaign, and then became his opponent on the analytical theory of heat proposed by Fourier.
Political Views
Initially, Fourier was an ardent Jacobist, but over time he became a moderate liberal.
It is believed that Fourier began to support the ideas of equality long before joining the committee, as evidenced by a letter from Fourier himself, written in June 1795 in prison, and joining the committee itself is associated with the desire to protect the republic from aggression from Belgium and the uprising in Vendee.
Jean Baptiste Joseph Fourier(French Jean Baptiste Joseph Fourier; March 21, 1768, Auxerre, France - May 16, 1830, Paris), French mathematician and physicist.
Biography
early years
Jean Baptiste Joseph Fourier was the 12th of 15 children in a tailor's family (the ninth in his father's second marriage). His father, Joseph Fourier, came from a family of shopkeepers from a small town in Lorraine. In the 16th and 17th centuries, Pierre Fourier (Saint Peter Fourier), the great-uncle of Jean Baptiste Fourier, was a well-known figure of the Counter-Reformation. His mother Edme died in 1777, when Fourier was nine years old, and his father died the same year. According to other sources, Fourier became an orphan at the age of eight.
At his first school, which was led by a church musician, Fourier showed success in studying French and Latin. At the age of 12, with the assistance of the Bishop of Auxerre, Fourier was sent to a military school at a Benedictine monastery. By the age of 13, Joseph became interested in mathematics, and at the age of 14 he mastered Bezout's six-volume Course of Mathematics. At the same time, he began collecting candle stubs in the school building so that he could study at night. In 1782-1783, Fourier received many prizes in rhetoric, mathematics, mechanics and singing. The long illness that followed may have been due to these intense activities.
At the age of 17, he dreamed of a military career and wanted to become an artilleryman or military engineer. Despite the support of school teachers and inspectors, Fourier was refused due to his humble origins. In 1787, Fourier entered the Abbey of Saint Benedict in the Loire, where he planned to be ordained. At the same time, the young man doubted his choice. In 1788, he sent his paper on algebra to Jean Etienne Montucla, but received no response. Fourier left the abbey in 1789 and went to the capital. In Paris, at the Royal Academy of Sciences, Fourier presented work on the numerical solution of equations of any degree.
During the Great Revolution
The revolution came before he could decide what to become - a monk, a military man or a mathematician. The revolutionary decree of October 1789 abolished religious vows, and soon the property of the church and monastic orders was confiscated. Fourier returned to Auxerre and began teaching mathematics, rhetoric, history and philosophy at the school from which he himself graduated. The Commissioner, who visited the school in October 1792, noted the liberal atmosphere of the classes and was dissatisfied only with the small number of Latin classes, which, at the request of the parents, gave way to mathematics classes
Until February 1793, Fourier was not involved in politics, despite the fact that the most militant regional branch of the Jacobin party was located in Auxerre. In 1793, a heated debate took place in Auxerre on the principles of separating people from the region at the request of the Convention. Fourier spoke at this debate and proposed a plan which was ultimately supported. In March 1793, Fourier received an offer to join the local oversight committee, which he accepted. By September of the same year, the committee, which was originally intended to suppress the counter-revolutionary activities of foreigners and travelers, became part of the revolutionary terror and was obliged to arrest “those who, by behavior, connections or words spoken or written, have shown themselves to be supporters of tyranny or federalism and enemies of freedom.” . Fourier, who did not want to participate in this, submitted a written resignation from the committee, which was rejected
On committee business, he went to the Loiret department. While passing by Orleans, he became involved in the conflict, speaking out in defense of the heads of several local families when the representative of the Convention made many arrests and intended to use a mobile guillotine. As a result, on October 29, 1793, his powers were revoked with the impossibility of obtaining them in the future, and Fourier returned to Auxerre in fear, where he continued to be a member of the local party branch and teach at school. Moreover, in June 1794 he became president of the revolutionary committee in Auxerre. After this, Fourier went to Paris for a meeting with Robespierre, which was not successful, since on July 4, immediately upon returning to Auxerre, he was arrested. He was already expecting the guillotine when, as a result of the coup of 9 Thermidor, Robespierre was arrested and executed, after which Fourier was released
French mathematician and physicist. Born into a tailor's family. At the age of 9, he lost both parents. The orphan was sent to a military school at a Benedictine monastery. In 1789 he came to Paris to present a work on the numerical solution of equations of any degree, but it was lost during the revolution. Fourier returned to Auxerre and began teaching at the school where he had previously studied.
In 1794 he entered the Normal School, organized by the Convention for the training of teachers. Soon the school was closed, but he managed to attract the attention of prominent scientists (Lagrange, Laplace and Monge). In 1795 - 1798 he taught at the Polytechnic School.
Participated together with other scientists in Napoleon's Egyptian campaign. He was the secretary of the Cairo Institute, founded by Napoleon. After the victory of England, in 1802 he was appointed prefect of the Isère department with headquarters in Grenoble, where he continued his scientific research in algebra, and worked actively in a new field of physics - the theory of heat. In 1808, Fourier received the title of baron and was awarded the Legion of Honor.
After Napoleon's defeat at Waterloo and the end of the "hundred days", he was removed from the post of prefect and moved to Paris. Here he worked for some time as director of the Bureau of Statistics, and thanks to the experience gained in Egypt, he raised this business to a height. In 1816, the Paris Academy of Sciences elected him as a member, but King Louis XVIII canceled the election. In 1816, the Academy of Sciences again elected him as a member, but this time the election was confirmed. Fourier became one of the most influential academicians and in 1822 he was elected secretary for life. In the same year he published the Analytical Theory of Heat (Théorie analytique de la chaleur). He died on May 16, 1830 in Paris.
Scientific achievements
He proved a theorem about the number of real roots of an algebraic equation lying between given limits (Fourier's theorem 1796).
He studied, independently of J. Muraile, the question of the conditions for the applicability of the method of numerical solution of equations developed by Isaac Newton (1818).
Monograph “Analytical Theory of Heat”, in which the derivation of the heat conduction equation in a solid body was given, and the development of methods for its integration under various boundary conditions. The Fourier method consisted of representing functions in the form of trigonometric series (Fourier series).
I found a formula for representing a function using an integral, which plays an important role in modern mathematics.
He proved that any arbitrarily drawn line, composed of segments of arcs of different curves, can be represented by a single analytical expression.
In 1823, independently of Oersted, he discovered the thermoelectric effect, showed that it has the property of superposition, and created a thermoelectric element.
