Equipped with his calculus Newton was able to explain with the help of the universal law of gravitation:
Astronomy which hitherto had been an empirical science transformed into a dynamical science. Laws of physics when expressed in mathematical terms result in differential equations. The two body problem of celestial mechanics results in a system of differential equations while the motion of a simple pendulum is a scalar second order ordinary differential equation.
In the presence of spherical or cylinderical symmetries, the study of these can be reduced to the study of important second order ordinary differential equations with variable coefficients such as the equations of Legendre and Bessel. We also look at other equations such as the Hermite's equation and the Airy's equation. This study must be preceded by a study of power series and their basic properties. Later we turn to Fourier series and Fourier transforms and integral representations of the solutions of the wave and heat equations. The course closes with a chapter on celestial mechanics on the inversion of the Kepler equation in terms of a Kaypten series specifically a Fourier sine series whose coefficients are in terms of Bessel functions.
Differential geometry is central to many parts of mathematics and sciences. Its indispensibility in general relativity is ofcourse well-known. Cartography is an ancient science and mathematical cartography received much impetus from Lambert and Flemish cartographers. One of the most important results in differential geometry - Theorema Egregium - of Gauss finds applications in cartography namely, the impossibility of making a perfect map. Within mathematics differential geometry interacts closely with complex analysis, topology and partial differential equations.
"Vollständige Erkenntiss der Natur einer analytischen Funktion muss auch die Einsieht den imaginären
Werthen des Arguments in sich Schliessen"
- Gauss in a letter to Bessel (1811)
These words augured the creation of a distinguished branch of analysis 14 years later by A. L. Cauchy, marked by the appearence of his memoir:
Mémoire sur les intégrales définies, prises entre des limites imaginaires (1825) Paris.
Autumn 2018: Differential Geometry & PDEs Seminar. We follow the monograph "Shape Variation and Optimization: A Geometrical Analysis" by Antoine Henrot and Michel Pierre. The book is