Augustin Louis Cauchy Josiah Blaisdell Valentin Moreau Rmy

Augustin Louis Cauchy Josiah Blaisdell, Valentin Moreau, Rémy Groux, Marion Saulcy

• I) The socio-economic context • II) His personal life • III) His mathematical contribution

I) The socio-economic context • 1789 : the Storming of the Bastille • 1792 – 1804 : French First Republic, separated in 3 parts : • The National Convention (1792 -1795) • the Directory (1795 -1799) • the Consulate (1799 -1804) 1804 -1815 : First French Empire

• 1815 -1830 : Bourbon Restoration • 1830 -1848 : July Monarchy • 1848 -1851 : Second French Republic • 1851 -1870 : Second French Empire

II) His personal life 1789 : birth of Augustin-Louis CAUCHY. 1805 : he placed second to the entrance examination (and he was admitted) to the Ecoles Polytechniques. 1807 : School for Bridges and roads. 1810 : graduated in civil engeneering with highest honor. 1812 : loose interest in engeneering, being attracted by abstract mathematics. 1815 : teach mathematics in Polytechniques. 1816 : Cauchy was appointed to take place in the Academy of science. 1818 : Cauchy maried Aloïse de BURE.

• 1830 : Cauchy leave the country • 1833 : Cauchy go to Prague to teach at the Duke of Bordeau (grandson of Charles X) • 1838 : come back in Paris. • 1839 -1843 : elected but not approved to take place in the Bureau des longitudes. • 1849 : teach mathematical astronomy at the Faculté des sciences. • 1857: death of August-Louis CAUCHY.

Contributions In Algebra • Cauchy Matrix • Cauchy Determinant of Cauchy Matrix • If the elements xs and ys are distinct then the determinant can be found quickly. • Group theory

Contributions in Analysis • Stressed the importance of rigor • “Infinitessimally small” quantities used to describe change. • Precursor to calculus. • Cauchy-Schwarz Triangle Inequality • Inner product and length • Cauchy Sequence • Used to construct the real line • Cours d‘Analyse • Cauchy-Riemann Conditions

Cauchy-Schwartz Inequality • Motivation: How can we go about assigning a length, or norm, to any vector in Rn? • Simple Examples:

Cauchy-Schwartz Inequality • Stating the inequality: • Or, using the dot product:

Cauchy-Schwarz Inequality • First show that the inequality holds when one of the vectors is the zero vector. • Notice that:

Conclusion