MATH 395 Special Topics OR PHYS 395

The major institution supporting research in mathematics and physics in the 18th century Europe was the French Académie des Sciences, founded in Paris in 1666 by Jean-Baptiste Colbert. Mathematicians and scientists from all over Europe, including Euler and Lagrange, entered regularly its prize competitions, hoping to win the challenges and the corresponding recognition, and ultimately to become members of the prestigious academy. In 1793, major institutions in France including the Academy of Sciences, were shut down by the Revolutionary Convention. Some scientists did not survive the reign of terror, but others, including Lagrange, did. In 1795 the Institute National des Sciences was established, and alongside L’Ecole Normale and L’École Polytechnique, institutions that would play a vital role in the development of mathematics and physics. Fourier, who started his studies at L’Ecole Normale attending Lagrange and Laplace lectures, put the foundations of an extremely rich area of research in mathematics that carries his name, Fourier Analysis. Lagrange together with Euler developed a new calculus, called variational calculus, which not only opened the door for new physics; it permitted the reformulation of existing theory, such as Newtonian mechanics, in a more elegant manner.
In this course, we will follow the paths in the history of mathematics leading to the development of variational calculus, differential equations, and approximation theory based on works of D’Alembert, Euler, Fourier, Lagrange, and Laplace. We will start the course at CNU, continue it in Paris, and complete it in Great Britain where the mathematics developed on the continent during the French Revolution made its way across the channel and opened the door for new applications. This transition relied heavily on the work of Mary Summerville, in Edinburgh, who progressed from solving math puzzles from the Ladies’ Diary to The Mechanism of the Heavens, ultimately playing a vital role by interpreting and translating Laplace’s works into English. On the pure physics side, aided by Summerville’s translations, Hamilton in Cambridge reformulated mechanics and Maxwell, who split his career between Edinburgh and Cambridge, unified electricity and magnetism in his famous set of partial differential equations. We will conclude the course in Great Britain with an introduction into these advances in physics.
Students will learn to apply course material to the larger world, to improve mathematical thinking and problem solving skills, to recognize the existence of cultural differences and acknowledge the perspectives and practices of others by exploring the monumental works of the giants of the 18th century scientists.