The Time of Lagrange, Fourier, and Cauchy 1790-1850

Thank You French Revolution! 1789- Napoleon Bonaparte becomes emperor of France and controls the military as well Believed officers should be trained in Mathematics and Science Founded Ecole Polytechnique to train officers Founded Ecole Normal to train the teachers of Mathematics and Sciences at Ecole Polytechnique Paris once again becomes center of math world during the 1st half of the 19th Century

http://upload. wikimedia http://upload.wikimedia.org/wikipedia/commons/thumb/e/e8/French_Revolution-1792-8-10.jpg/800px-French_Revolution-1792-8-10.jpg

http://content. answers http://content.answers.com/main/content/wp/en-commons/thumb/d/da/350px-Prise_de_la_Bastille.jpg Napoleon Bonaparte http://members.tripod.com/mr_sedivy/france/napolean.jpg

Joseph-Louis Lagrange 1736-1813

Joseph-Louis Lagrange 1736-1813 Organized his works into influential textbooks that played a significant role in the spread of calculus results Called for a greater level of rigor in calculus Used power series concept as the basic foundation in the development of calculus. (he developed this idea in his Theorie des fonctions analytiques ) Cauchy based calculus concepts and results on a careful foundation, but he choose the limit concept instead of the power series for this purpose.

Joseph-Louis Lagrange 1736-1813 Married his cousin Under Napoleon he was a member of the senate and made a count “I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable.” “When we ask advice, we are usually looking for an accomplice.”

Joseph B. Fourier 1768-1830

Joseph B. Fourier 1768-1830 His main book Theorie analytique de la chaleur presented a mathematical solution to the partial differential equation for heat flow. He solved it by using infinite series of trigonometric functions Developed the Fourier Series- any periodic function can be formed from the infinite sum of sines and cosines- can also be found in his book Theorie analytique de la chaleur Up until Fourier series, Power series were used to represent trigonometric and exponential functions and their inverses but they could not represent discontinuous functions- Fourier series provided a way to represent discontinuous functions which opened doors for calculus

Discoveries that occurred because of Fourier Series George Cantor worked on the convergence problem for Fourier series from this he began his work on sets which was very influential Cantors ideas on sets revolutionized thinking about the infinite

Successive approximations to common functions using Fourier series http://mathworld.wolfram.com/FourierSeries.html

Joseph B. Fourier 1768-1830 Mother died when he was 9 and his father died the following year. Was going to be a priest but his love affair with math consumed him. He was a professor of Analysis at the Ecole Polytechnique. Advisor was Lagrange Renowned as an outstanding lecturer “Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.”

Augustin-Louis Cauchy 1789-1857

Augustin-Louis Cauchy 1789-1857 Home schooled by his father until he was 13 Entered Ecole Centrale du Pantheon to study language upon the advice of Lagrange (family friend) Entered Ecole Polytechnique at age 16 Graduated at age 18 and worked as civil engineer

Augustin-Louis Cauchy 1789-1857 He was the first person to show how the limit concept could serve as the foundation for calculus Based the following on the concept of a limit: continuity, derivative, definite integral, sequences and series Along with Niels Abel- credited for developing many of the standard convergence tests for infinite series Did not see need for concept of uniform convergence One of his theorems stated that the limit of a convergent sequence of continuous functions must be a continuous function but this was incorrect (Abel) and he was reluctant to accept that; need concept of uniform convergence to prove such a theorem (1848 Paul Seidel)

Augustin-Louis Cauchy 1789-1857 Went into voluntary exile when the King was removed from office after the 1830 revolution He refused to take any oaths of allegiance that were required in order to teach at the university Misplaced the papers of young mathematicians who had sent their work to The Paris Academy for evaluation. Niels Abel Younger Mathematician (1802-1829) sent a letter to a friend who described Cauchy as a fool, extremely Catholic and bigoted

Carl Friedrich Gauss 1777-1855 www.jfinternational.com/.../images/gauss8.gif http://www.rare-earth-magnets.com/images2/johann_carl_friedrich_gauss.jpg

Carl Friedrich Gauss 1777-1855 His main contributions were in the fields of number theory, geometry, and astronomy He discovered a proof that the regular polygon of 17 sides could be constructed using only a compass and a straightedge. Wrote a famous book Disquisitiones arithmetica which launched the modern theory of numbers One real contribution to the development of real analysis (1813)- he was the first to develop a rigorous treatment of convergence when he discussed convergence of the hypergeometric series (he did not carry these efforts further) Many disgruntled mathematicians announced their work only to have Gauss produce evidence that he had developed the idea years before.

Bernard Bolzano 1781-1848 Developed proof for the intermediate value theorem from analysis Developed some properties of limit points for a sequence from analysis Bolzano-Weierstrass theorems Most of his results were published after his death

Prague, capital of Czech Republic Bernard Bolzano http://www.easyprague.cz/eecera2007/download/images/prague-bridges.jpg http://www.dcs.warwick.ac.uk/bolzano/images/I.15zoom.gif Prague, capital of Czech Republic